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rata 

0 


telure. 


□ 


32X 


1 

2 

3 

1 

2 

3 

4 

5 

6 

A^ 


^■^■■■■11 


AN 


ELILMHNTARV    TRHATISE 


ANALYTIC  3IKrilANICS 


WITH    NUMEROUS     EXAiMPLES. 


BY 


K 


p]DWAHD  A:'  HOWSER.  LL.D.. 

PROFESSOR     OR     MATHEMATICS     AND     KN<ilM:HRm(;     IN     KUTGERS     COLLECB. 


NINTH    EDITION. 


NEW    YORK: 
D.    VAN    NOSTRAND    COMPANY 

23  MuuRAY  St.  and  27  Warren  St. 
1890. 


' .  l"^ 


Q^CA- 


coPYRianT,  JHHi,,  iir  K  a.  bowser. 


53 

JAN  24  1946 
•wlal  Rm«N  MvMm 


J 


PREFACE. 


-*-»■♦- 


r  pHE  preHent  work  on  Analytic  Mechanics  or  Dynamics  is  (lewgnea 
as  a  text  booli  lor  tlio  students  r)f  ScientiKc  t^cliools  and  Col- 
leges, who  iiave  received  training  iu  the  elements  of  Analytic  Geome- 
try and  tlie  Calculus. 

Dynamics  is  hero  used  in  its  true  senso  as  the  science  of  furce 
The  tendency  anion<r  the  l)est  and  most  logical  writers  of  the  presrn', 
day  ajjjjears  to  be  to  us  ■  this  term  for  the  science  of  Anal  vlic  Me- 
chanics, while  the  branch  formerly  called  Dynamics  is  now  termed 
Kinetics. 

The  treatise  is  intended  es|)ecially  for  beginners  in  this  branch  of 
science.  It  involves  the  use  of  Analytic  (jetmietry  and  the  Calculus. 
The  analytic  method  has  been  chiefly  adhered  to,  ns  beinp  better 
adapted  to  the  treatment  of  the  subject,  more  general  in  its  applica- 
tion and  more  fruitful  In  results  tluui  the  geometric  method;  anii  yet 
where  a  geometric  proof  seemed  jjreferable  it  has  been  introducrnl. 

The  aim  has  been  to  make  every  principle  clear  and  intelligible, 
to  develop  the  different  theories  witli  simplicity,  and  to  exi)lain  f'llly 
the  meaning  and  use  of  the  various  amilytic  expressions  in  which  the 
principles  are  embodied. 

The  book  consists  of  three  parts.  Part  I,  with  the  exception  of  a 
preliminary  chapter  devoted  to  definitions  and  fundaiiiental  princi- 
jih'S,  is  entirely  given  to  St/iticK. 

Part    II   is  occupied   with   Kinematics,  and   the  ])rinciples  of  <]\\n 
i<up(»innt  briiuch  of  mathenuitics  are  so  tri-ated  that  the  student  may 
,iite>  \ipon  the  study  of  Kinetics  with  clear  notions  of  motion,  veloc 
itv  mid  accehratioii.     Part  III  treats  of  the  Kluetics  of  a  particle  and 
of  rigid  bodies. 


IV 


PREFA  CE. 


In  this  arrangement  of  the  work,  with  the  exception  of  Kine- 
luatica,  I  have  followed  the  plan  usually  adopted,  and  made  the 
subject  of  Statics  precede  that  of  Kinetics. 

For  the  attainment  of  that  pfrtisp  of  jmnciiiles  which  it  is  the 
special  aim  of  the  lK)ok  to  impart,  numerous  examjjles  are  given  at 
the  ends  of  the  chapters.  The  greater  part  of  them  will  present  no 
serious  difficulty  to  the  student,  while  n  few  may  tax  his  best 
eflTorts. 

In  preparing  this  book  I  have  availed  myself  of  the  writings  of 
many  of  the  best  authors.  The  chief  sources  from  which  I  have 
derived  assistance  are  the  treatises  of  Price,  Mincliin,  Todhunter, 
Pratt,  Routh,  Thomson  and  Tait,  Tail  and  Steele,  VVeisbach,  Ventu- 
roll,  Wilson,  Browne,  Gregory,  llankine,  Bouchiirlnt,  Pirie,  Lagrange, 
and  La  Place,  while  many  valuable  hints  as  well  as  exampl  ;8  have  been 
obtained  from  the  works  of  Smith,  Wood,  Bartlett,  Yoiing,  Moseley, 
Tate,  Miignus,  Goodev.',  Parkinson,  Olmsted,  Garnett,  Renwick,  Bot- 
tomley,  Morin,  Twisden,  Whewell,  (ialbraith,  Ball,  Dana,  Byrne,  the 
Encyclopedia  Britannica,  and  the  Mathematical  Visitor. 

I  have  again  to  thank  my  old  pupil,  Mr.  R.  W.  Prentiss,  of  the 
Nautical  Almanac  Office,  and  formerlv  Fellow  in  Mathematics  at  the 
Johns  Hopkins  Univ.'rsity,  lor  reading  the  MS.  oud  for  valuable  sug- 
gestions. Several  others  also  of  my  friends  have  kindly  assisted  me 
by  correcting  proof-sheets  and  verifying  copy  and  fonnulro. 

E.  A.  B. 
RuTOEits  College,  i 

New  Brunswick,  N.  J.,  June,  1884. ' 


TABLE   OF   CONTENTS 


PART    I. 


CHAPTER    I. 


1. 
2. 

3. 

4. 

5. 

6. 

7. 

H. 

!». 
10. 
11 
12. 
18. 
14. 
15. 
10. 
17. 
18. 
11). 
30. 
21. 
22. 

2a. 


FIRST    PRINCIPLES. 

Definitions — Statics,  Kinetics  and  Kinematics. , 
Matter 


PAOB 
1 

a 

Inertia 2 

Body,  Space  and  Time 3 

Rest  and  Motion 3 

Velocity 3 

Formulae  for  Velocity 4 

Acceleration 4 

Measure  of  Acceleration 5 

Oeonietric  Representation  of  Velocity  and  Acceleration 6 

.  MnsH " 

Momentum 7 

Cluinge  of  Momentum ^ 

Forc(! •■  •  •  •  8 

Static  Measure  of  Force 8 

Action  and  Ht'action 9 

Metliod  of  Comparing  Forces 9 

Keprescntatiim  of  Forces 10 

McUKure  of  Arceleratin)r  Forces 10 

Kinetic  Measure  of  Force 11 

Al)8olute  or  Kinetic  I'nit  of  Force 1!' 

Tliree  Ways  of  Measuring  Force ■  ■  ■  14 

Meaning  of  </  in  Dynamics 15 


»J'^*>i°»««i^i!,'-«6WVS*B»t . 


mm 


VI 


VUNTEJ^iTS. 


*"'■'•  PAGE 

24.  Gravitation  Units  of  Force  and  Mass 10 

~o    Gravitation  and  Aljsoluto  Measure i; 

E^iamples .  ,    13 


STATICS     (REST). 


CHAPTER    II. 

THE      COMPOSITION      AND      KKSOLUTIOX      OF     COXCURRINO 
FORCES — CONDITIONS    OF    EQUILIBRIUM. 

26.  Problem  of  Statics 21 

27.  Concurring  and  Conspiring  Forces 21 

28.  CV-m position  of  Conspiring  Forces 23 

29.  Composition  of  Velocities 28 

80.  Composition  of  Forces 24 

31.  'I'riangle  of  Forces 25 

32.  Three  Concurring  Forces  in  Eiiuilibriuni 2G 

33.  The  Polygon  of  Forces 27 

34.  Parallelopiped  of  Forces •. 28 

35.  KesohitixMi  of  Forces 30 

30.  Magnitude  an<l  Direction  of  Hesultunt 31 

37.  Conditloiis  of  Kquiiil)rium 32 

33.   Resultant  of  Concurring  Forces  in  Sjiace 34 

39.  Equilibrium  of  Concurring  Forces  in  Space 35 

40.  Tension  of  a  String 35 

41.  Equilibrium  of  Concurring  Forces  on  a  Smooth  Plane 39 

42.  Equilibrium  of  Concurring  Forces  on  a  Smooth  Surface 41 

Examples 45 


48. 
49. 
50. 
51. 
52. 
5:{. 
.54. 
55 
56. 
57. 
58. 
59. 
60. 
«1. 
62. 
<i3. 
64. 
65. 


CHAPTER    III. 

COMPOSITION    AND     RESOLUTION    OF   FOKCES    ACTING    ON  A 

HKill)   liODY. 

4:!.   A  Rigid  Body ...  >57 

1 1.  Transmissil)ility  of  Force 57 

45.  Resultant  of  Two  Parallel  Forces gg 

46.  Moment  of  a  Force  (jg 


PAOB 

ART 

....     IG 

47. 

....     17 

48. 

....     13 

49. 

50. 

51. 

52. 

5;i. 

.54. 

55 

56. 

L'RRINfi 

57. 

58. 

59. 

....     21 

60. 

....     21 

61. 

. . . .     22 

(52. 

. . . .     28 

63. 

. . . .     24 

64. 

. . . .    2n 

65. 

....     20 

....     27 

....     28 

...     31 

...     32 

. . . .     34 

. . .     ',t~t 
...     35 
. . .     3!) 
...     41 
...     45 

60. 
07. 
68. 
69. 
70. 

71. 

72. 

73. 

ON  A 

74. 

7."). 

70. 

...     57 

t  1  . 

...     57 

7S. 

...     58 

7!). 

...     60 

80. 

<  ONTEiWrs. 


Til 


Signs  of  Mornents 01 

Uoomotric  Hcitrcscntation  of  a  Moment . 61 

Two  K(iual  and  Opposite  Farttlli^l  Forces 01 

Moment  of  a  Coiiiile 02 

Etl'ect  of  a  Couple  on  a  Rigid  IJotly 03 

Effect  of  Transferriiig  Couple  to  Parallel  Plane  not  altered. . .  04 

A  Couple  rc'iilaci'd  by  another  Couple 61 

A  Force  and  a  <  'oi:i)le 65 

Resultant  of  any  number  of  Cou])li's 60 

Resultant  of  Two  l'oui)les 07 

Varignon's  Theorem  of  Moments 69 

Varignon's  Theorem  for  Parallel  Forces 71 

Centre  of  Parallel  Forces 71 

Equilibrium  of  a  Rigid  Hody  under  Parallel  Forc(>s 74 

E(iuilibriuin  of  a  Rigid  Body  unde".  Forces  in  any  Direction. .  75 

E(iuilibrium  under  Three  Forces 77 

Centre  of  Parallel  Forces  in  Different  Planes 85 

Eiiuilibrium  of  Parallel  Forces  in  Space 80 

Eiiiiilibrium  of  Forces  acting  in  any  Direction  in  Sjmce 88 

Examples 90 


CHAPTER    IV. 

CKXTRE   OF   GRAVITY    (CENTRE   OF   MASS). 

Centre  of  Gravity 100 

Planes  of  Symmetry — Axes  of  Symmetry 101 

Body  Suspended  from  a  Point 101 

Body  Supported  on  a  Stirt'ace 102 

DilTerent  Kinds  of  E(iuilibrium 102 

Centre  of  (iravity  of  Two  Masses 103 

Centre  of  (iravity  of  Part  of  a  Body 103 

Centre  of  ( havity  of  a  Triangle 103 

Centre  of  Gravity  of  a  Triangular  Pyramid 105 

Centre  of  Gravity  of  a  Cone 100 

Centre  of  Gravity  of  Frustum  of  Pyramid 107 

Investigi'tions  involving  Integration 109 

Centre  of*  iravity  of  the  Arc  of  a  Curve 110 

Centre  of  Giavity  of  a  Plane  Area 115 

Polar  Elements  of  a  Plane  .\rea 1 18 


Vlll  coy  TENTS. 

ABT.  PAGE 

Hi.  Doiihlo  Tntoj^TRtion — Polar  Formulen 120 

83.  Double  Integration — l{ectangnl«r  Formulr 122 

8;$.  Centre  of  Gravity  of  a  Surface  of  Revolution 12!} 

s4.  Centre  of  (iravity  of  any  Curved  Surface 120 

85.  vVnlre  of  Gravity  of  a  Solid  of  Revolution 127 

8<5.  Polar  Formulae 130 

87.  Centre  of  Gravity  of  any  Solid 1;J1 

H8.  Polar  Elements  ot  Mass 133 

89.  Special  Methods 13« 

90.  'I'heorems  of  Pappus 138 

Examples 140 


CHAPTER    V. 

FRICTIOX. 

91.  Friction 149 

92.  Laws  of  Friction 15(1 

93.  Magnitudes  of  Coefflcients  of  Friction 152 

94.  Angle  of  Friction ".....  152 

95.  Reaction  of  a  Rough  Curve  or  Surface 153 

90.  Friction  on  an  Inclined  Plane 154 

9'i.  Friction  on  a  Double  Inclined  Plane 156 

98    Friction  on  Two  Inclined  Planes 15!) 

99.  Friction  of  a  Trunnion 159 

100.  Friction  of  a  Pivot 100 

Examples 162 


CHAPTER    VI. 

TIIK   PKINCIPLK   OF   VIKTl'Al,    V  KLOCITIES. 

101.  Virtual  Velocity "in 

102    Princijile  of  Virtual  Velocities ir>7 

103.  Xiiture  of  the  Displacement 1(!!) 

104.  E(|uation  of  Virtual  Moments 101) 

105.  System  of  Particles  Rigidly  Connected 170 

Examples IT'? 


UCT. 

106.  Fun 

107.  Mec 

108.  Sim 

109.  The 

110.  Equ 

111.  The 
113.  Chi. 

113.  The 

114.  To  ( 

115.  The 

116.  Eq<i 

117.  To.) 

118.  Reli 

119.  Reli 

120.  The 

121.  The 

122.  The 

123.  Firs 

124.  Sec. 

125.  Thii 
136.  The 

127.  Mec 

128.  The 

129.  Reli 
129«.  Pr 

Exn 


THE 

Fl 

130. 

E.,. 

131. 

To 

132. 

Cor 

133. 

Th 

VA'An 

.  .\ 

Ex 

ir.7 

Hi!) 
17(1 
179 


lO^fTHA'TS. 


U 


PAGE 

120 

122 

12:5 

120 

ABT. 

127 

106. 

130 

107. 

i;}i 

108. 

133 

109. 

136 

110. 

138 

111. 

140 

112. 

113. 

114. 

115. 

116. 

. 

117. 

118. 

119. 

149 

120. 

150 

121. 

152 

122. 

152 

123. 

153 

124. 

154 

125. 

156 

126. 

15!) 

127. 

159 

128. 

160 

129. 

162 

129« 

CHAPTER    VII. 

MACHINES. 

FAOI 

Functions  of  a  Miicliine 177 

Mechanical  Advantage 178 

Simple  Machines 180 

Tlie  Lever 181 

Equilibrium  of  the  Lever 181 

The  Common  Balance.   184 

Chief  Requisites  of  a  Good  Balance 186 

The  Steelyard 188 

To  Graduate  the  Common  Steelyard 188 

The  Wheel  and  Axle 190 

Equilibriuiu  of  the  Wheel  and  Axle 190 

Toothed  Wheels 192 

Relation  of  Power  and  Weight  in  Toothed  Wheels 193 

Relati(m  of  Power  to  W^Mght  in  a  Train  of  Wheels 194 

The  Inclined  Plane 196 

The  Pulley 197 

Tiie  Simi)ie  Movable  Pulley 198 

First  System  of  Pulleys 198 

Sec<md  System  of  Pulleys 200 

Third  System  of  Pulleys 201 

The  Wedge 202 

Mechanical  Advantage  of  the  Wedge 202 

The  Screw 204 

Relation  between  Power  and  Weight  in  the  Screw 204 

J.  Prony's  Differential  Screw 206 

Examples 807 


CHAPTER    VIII. 

THE   FUNICULAR   POLYGON— THE   CATENARY — ATTRACTION 

130.  Eiiuilibrium  of  the  Funicular  Polygon 216 

131.  To  Construrt  the  Funicular  Polygon 218 

132.  Cord  SupiMjrtiiig  a  Load  L'niformly  Distributed 219 

133.  The  Common  Catemiry — Its  E(|Uution 221 

133((.  Attraction  of  a  Spher!<'iil  Shell 220 

Examples 228 


.  w»SBSS^».^|ISte , , 


COyTENTH. 


PART    II. 
KINEMATICS     (MOTION). 


CHAPTER    I. 

RECTILINEAR     MOTION. 

ART.  PAOB 

134.  Definitions — Velocity 231 

135.  Accfluriition 233 

130.  Relation  hctwoon  Space  and  Time  when  Acceleration  =  0.. .  233 

137.  Relation  when  the  Apcelerutitin  is  Constant 234 

138.  Relation  when  Acceleration  varies  us  the  Tinu' 235 

139.  Relation  when  Acceleration  Varies  as  the  Distance 235 

140.  Equations  of  Motion  tor  Faliiiifi;  Rodies 237 

141.  Particle  Projected  W'rticully  Upwards 230 

142.  Compositions  of  Velocities 242 

143.  Resolution  of  Veldciiies 243 

144.  Motion  on  an  Inclined  Plane 245 

145.  Times  of  Descent  down  Chords  of  a  Circle 247 

:246.  The  Straifrht  Line  of  (Quickest  Descent 248 

Examples 249 


ART. 

lot). 
Vu. 
158. 
159. 
1(50.  Anjl 
l(il.  Aci 
102.  Ace 
l(i:l  Wl 
1()4.  Wli 
Exi 


CIlAl'TKR     II. 

C  U  H  V  I  1,  I  N  K  A  R     M  0  T  I  O  N  . 

147.  Remarks  on  Curvilinear  Motion 

148.  Composition  of  Uniform  Velocity  and  Acccderalion 

149.  Composiliun  and   Hesolution  of  .Xccelinition. 

Examples 

150.  Motion  of  Projectiles  in  Vacuo 

151.  The  Path  of  a  Particle  in  Vai'uo  is  a  Parabola 

152.  The  Parameter — Ranp-— (freatest  llei^'lit — lleiirht  of  Direc 

trix 

153.  V'el'icity  of  a  Particle  at  any  ))oint  of  its  Path 

154    Time  of  Flif,'lit  alontr  a  Ilori/oiitiil  Plane 

155.  Point  at  which  a  Projectile  will  Strike  an  Inclined  Plane. . . 


S58 
258 
'■ly.) 
2(il 
2:;({ 
2f.O 

2(17 
2(1!> 
2(19 
270 


171. 
172. 


175 
17(1. 
17r. 


CONTENTS. 


XI 


PAOB 

.  231 

.  233 

.  233 

.  234 

.  235 

.  235 

237 

239 

242 

243 

245 

247 

248 

249 


25S 
258 
'J')!) 
2(11 

2;i(i 
2('0 

2((7 
2(i!> 
2(!it 
27(J 


iRT. 

150. 
157. 
158. 
1,59. 
1(50. 
101. 
102. 
103. 
104. 


PAQB 

Projectinn  for  Orentest  Range  on  a  Given  Plane 270 

The  Elevation  that  the  l':iiticli-  may  pass  a  (iiven  Point   .  .     271 

Second  Method  of  Finding  E(iii»tion  of  Trajectory :>7i! 

Velocity  of  Discluu-jfr  of  Balis  and  ShcUn 274 

Angnlar  Velocity  ;iiid  Angnlar  Acceleration 275 

Accelenitioiis  Aloiiff  and  Perpendicular  to  Radius  Vector. . . .  278 

Accelerations  Along  and  Per|)eiidicular  to  Tungent 279 

When  Acceleration  Perpendicular  to  Kadius  Vector  is  zero. .   281 

When  Angular  Velocity  is  Constn-t 283 

Examples 3y4 


PART    III. 
KINETICS     (MOTION    AND    FORCE), 


CHAPTER    I. 

LAWS     OF     MOTION — MOTION'     l-NDER     THK     ACTION     OF     A 
VAUIAin>K   FOUri;— MOTION   IN   A    RESISTING   .MEDIUM. 

105    Peflnitions 289 

160.  Newton's  Lawn  of  Motion 289 

107.  Remarks  on  Law  I  290 

lOS.  Hennuks  on  Law  II 291 

109.  Remarks  on  Law  III 294 

170    Two  Laws  of  Motion  in  the  I-^'ench  Treatises 295 

171.  Motion  of  Particle  under  an  Attractive  Force 295 

172.  Motion  under  the  action  of  a  Variable  R('|)ulsive  Force 298 

173.  Motion  under  the  action  of  an  Attractive  Force 299 

174.  Velocity  aci|uired  in  Fallinir  tlirougli  a  (ireat  Height 300 

175    Motion  in  a  IJesinting  Medium 302 

I  TO.  Motion  in  the  .\ir  against  the  .\ction  of  Gravity 304 

177.  Motion  of  a  Projectile  in  a  Resisting  Medium 307 

178    Motion  against  the  Krsistance  of  the  .\tinosphere 308 

179.  ^foMon  in  tlu>  .\tmos|ihere  under  a  small  .\ngle  of  IClevatioii  312 

Ex:ini|iles 313 


ifitaita 


XI 1 


CONTENTS. 


CHAPTER    II. 

CENTRAL   FORCES. 

180.  Definitions •{■jl 

181.  A  Particle  under  tlie  Action  of  a  Central  Attraction 'd'i\ 

182.  The  Sectorial  Area  Swept  over  by  the  Hadius  Vector 'A'i't 

18;f.  Velocity  of  Particle  at  any  Point  of  its  Orbit 82.'j 

184.  Orbit  when  Attraction  as  the  Inverse  Square  of  Distance. . .  32!l 

185.  Suppose  the  Orbit  to  be  an  Ellipse Jj3ij 

186.  Kei)ler's  Laws ij;^,;; 

187.  Nature  of  the  Force  which  acts  upon  the  Planetary  System .  335 
Examples ggg 


188. 
18!). 
190. 
101. 
192. 
103. 
1!I4. 
195. 
lOB. 
197. 
198. 
199. 
200. 
301. 


CHAPTER    III. 

CONSTRAINED     MOTION 

Definitions 345 

Kinetic  Energy  or  Vis  Viva  —Work 345 

To  Find  the  Heaction  of  the  Constraining  Curve 348 

Point  where  Particle  will  leave  (lonstr.iining  Curve.  .• 349 

Constrained  Motion  Under  Action  of  Oravity 350 

Motion  on  u  Circular  Arc  in  a  Vertical  Plane 850 

Tlie  Simple  Peixluluin 353 

Kelation  of  Time,  Length,  and  Force  of  (Jravity 8,')3 

Height  of  Mountain  Determined  with  I'endulum 354 

Dejith  of  Mine  Determined  with  Pendulum 355 

Centripetal  and  Centrifugal  Forces. ;J5(J 

The  Centrifugal  Force  at  the  Ecpuitor SDS 

Centrifugal  Force  at  Difl'erent  Latitudes 359 

The  Conical  Pendulum— 'l"he  (foveriior.         ;j6i 

Examples 3^3 


217. 


CHAPTER    IV. 

IMP.\(T. 

202.  An  Iiu|)ulsive  ["'orce ;)~q 

203.  Imi)act  or  Collision. jj^i 

20 1,   Direct  and  ( 'enl  ml    lnipa<  t y^-o 

205.  Elasticity  of  Hodies— t'oettioieni  of  Hestltution 373 


CONTENTS. 


Xlll 


PAOB 

321 

3','1 

M5 

325 

ce...  32!) 

333 

335 

Item.  335 
838 


345 
345 
348 
34» 
350 
850 
352 
853 
354 
355 
350 
358 
359 
361 
862 


370 
371 

373 


^'^-  PACK 

200.  Direct  Impact  of  Inelnstic  Bodies 374 

207.  Direct  linpiict  of  Elastic  Bodiest 375 

208.  Lows  of  Iviuetic  Energy  in  hnpact  of  Bodies 378 

209.  ()l)ii{iue  Inijmct  of  Bodies. ...  380 

210.  Obll()ue  Iini'act  of  Two  Smootli  Spheres 382 

Examples 3B3 

CHAPTER    V. 

WOKK   AND   ENERGY. 

211.  Definition  and  Meiisiire  of  Worli 3Si) 

212.  (Jenernl  ('a.«e  of  Worii  done  by  a  Force 390 

213.  Work  on  an  Inclined  Plane 391 

Exam|.les 3()3 

214.  Horse  Power 89r) 

215.  Work  of  Kaisinpr  a  System  of  Weights 3!)G 

ExanijdeH 397 

210.  Modulus  of  a  Machine 400 

Examples 40) 

217.  Kinetic  and  Potential  Energy — Stored  Work 404 

Exnmi)le8 406 

218.  Kinetic  Energy  of  a  Rigid  Body  Revolving  round  an  Axis. . .  408 

219.  Force  of  a  Blow 411 

220.  Work  of  a  Water  Full 412 

221.  The  Duty  of  an  Enjrine 414 

222.  Work  of  a  Variable  Force 415 

223.  Simpson's  Rule 415 

Examples 417 

CHAPTER    VI. 

MOMENT  OF    INERTIA. 

224.  Moment  of  Inertia 429 

i.'25.  Moments  of  Inertia  relative  to  Parallel  Axes  or  Planes 432 

226.  Riulins  of  Oyration 4;(,{ 

227.  Poliir  Moment  of  Inertia 4;J0 

228.  Miimeni  of  Inertia  of  11  Solid  of  Uevolution 4:i7 

229.  Moment  of  Inertia  iiboiit   Axis  Perpendicnliir  to  (Jioinetric 

Axis .l;t8 

230.  Moment  of  hurtiii  of  \';uioiis  Solid  U(ldil■^^ .|4(l 

231.  Moment  of  Inertia  of  a  Laininu  with  respect  to  any  .Vxis.  ...  441 


^Mi 


XIV 


VONTBNTS. 


ART.  J"*''" 

2;$2.  Priiici|wl  Axes  of  a  Body 14  M 

233.  Prodiuts  of  Inertia 4 Ki 

Exuiuplos -WT 

CHAPTER     VII. 

HOTATOUY    MOTION. 


234.  Impressed  and  Effective  Forces 

235.  D'Aleiiiljert's  Princii)le 

230.  notation  of  a  Rifrid  Body  al  out  a  Fixed  Axis 

237.  Tlie  ("oiiipound  Pendulum 

238.  Lenjith  of  Second's  Pendulum  Di'tennined  Experimentally.. 

239.  Motion  of  a  Hody  when  Unconsiraimd 

^40.  CentH!  of  Percussion— Axis  of  Siiontancons  Rotation 

241.  Principal  Radius  of  (iyration  Determined  P  actically 

242.  The  Hallistic  Pendulum 

243.  Motion  of  a  Body  al)out  a  Horizontal  Axle  through  its  Centre 

244.  Motion  of  a  Wheel  and  Axle 

24.").  Motion  of  a  Rigid  Body  about  a  Verticid  Axis 

240.  Body  Rolling  down  an  Inilineil  Plane 

247.  Falling  Body  under  an  hni>ulse  not  through  its  Centre 

Examples 


4.11 

4r)2 

4.54 

4.-)7 

4(;2 

4(il 

4(14 

407 

40H 

470 

471 

472 

473 

47.-. 

477 


OirAPTHU     VIII. 

MOTION    OF    A    SYSTKM    OF    UIOIO    HODIKS    I\   SI'AOK. 

248.  Equations  of  Motion  ohtiiined  by  D'Alenihert's  Principle   .  .  . 

249.  Independence  of  the  Motions  <if  Translnlion  and  Rotation. .  . 

250.  Principle  of  the  Conservation  of  the  Centre  of  (Jravily 

251.  Principli'  of  the  Conservation  of  Areas 

252.  Conservation  of  Vis  Viva  or  Energy 

253.  Coniposiiion  of  Hotiitions 

2.54    Motion  of  a  Rigid  Body  referred  to  Fixid  Axes 

255.  Axis  of  Instantaneous  Rotation 

2.50.  Angular  Velocity  about  Axis  of  Instantaneous  Rotatimi 

257.  Eulei's  Ivpialions • 

25S.  Motion  alxiut  a  Principal  Axis  through  Centre  of  (irnvity.  .  . 
■.'.■)it.   V(docity  about  a  Principal    Axis  when   Accelerating  Forces 

0    

■iOI).  'I'hi'  IiitfLiiiil  of  iMiler's  Eiiuiitioiis 

Examples 


481 
4H2 
485 
4H(i 

48S 

49;i 

494 
495 
490 
497 

499 

.501 
5()."i 


ANALYTIC    MECHANICS. 


PART    I 


C  H  A  P  T  F.  R     I . 

FIRST     PRINCIPLES. 

1.  Definitions. —.lH'^/////r  M''cii(inirn  or  D;/namics  is 
the  fjok'iu'i'  wlik-li  treats  of  the  cqiiilibriuiii  aiul  niot.on 
of  iHxlies  uii'ler  tlie  net  ion  of  force.  It  is  aeeordiiigly 
(livideil  into  two  luirts,  Shi/ics  nnd  Kineliu. 

Slatim  treats  of  tlie  e(|uilil)riiiiii  (jf  Ixjdies,  and  the  CDiidi- 
tions  governiniT  the  f.ircis  wiiieh  pi'oihiee  it. 

KiHclicK  treats- of  tlie  motion  of  bodies,  and  the  hiws  of 
the  forces  whicii  proiluce  it. 

The  consideration  liiat  the  properties  of  motion,  velocity, 
and  disi)lacenient  may  l)e  tri'ated  ai>art  from  tlie  jiarticiilar 
forces  prodncing  tliein  and  independently  of  the  bodies  sub- 
ject to  them,  has  given  rise  to  an  auxiliary  branch  of  Dyna- 
mics called  KiiH'iiKi/icx* 

Although  Kiuenndics  may  not  be  regarded  as  properly 
included  under  Dynamics,  yet  this  branch  of  science  is  so 
imi»orlant  anil  useful,  and  its  ii|ti)IIiMtion  to  Dynamics  so 
immediate,  that  a  portion  of  this  work  is  devoted  to  its 
treatment. 


*  This  imnu'  win'  i;iviii  l)j  AmiiSro. 


* 


2  MATTER,   INERTIA,   BODY,   MOTION,   ETC. 

Kiiu'iiiatics  is  tlic  scienee  of  pure  motion,  without  refer- 
ence  to   mutter  or  force.     It  treats  of  the  properties  of 
motion    without   regard  to  what  is  moving  or  how  it  i*; 
moved.     It  is  an  extension  of  jiure  geometry  by  introduc- 
,  ing  the  idea  of  time,  and  the  consecpient  idea  of  velocity. 

2.  Matter. — Ma/icr  is  that  wliicli  can  be  perceived  bv 
the  senses,  and  which  can  transmit,  and  be  acted  upon  bv 
force.     It  has  extension,  resistance,  and  impenetrability. 

A  di'finition  of  iimtter  which  woiilil  satisfy  tlie  nietaphysicinn  is 
not  reqiiirwl  for  tliis  work.  It  is  suflScient  for  us  to  conceive  of  it  as 
cai)iihle  of  receiving  and  transmitting  force  ;  because  it  is  In  this 
a.spect  only  that  it  is  of  importance  in  the  present  treatise. 

3.  Inertia.— By  Inert ia  is  meant  that  property  of  mat- 
ter by  which  it  remains  in  its  state  of  rest  or  uniform 
motion  in  a  right  line  unless  acted  ui)on  by  force.  Inertia 
expresses  the  fact  that  a  body  cannot  of  itself  change  its 
condition  of  rest  or  motion.  It  follows  that  if  a  body 
change  its  state  from  rest  to  motion  or  from  motion  to  rest, 
or  if  it  ciiange  its  direction  from  the  natural  rectilinear 
path,  it  must  iiave  been  inlluenced  by  some  external  cause. 

4.  Body.  Space,  and  Time.— -1  Body  is  a  portion  of 

matter  limited  in  ev.   y  direction,  and  is  consequently  of  a 
determinate  form  and  volume. 

A  Riijiil  Bodji  is  one  in  which  the  relative  jjositions  of 
its  particles  remain  unchanged  by  the  action  of  forces. 

A  Particle  is  a  body  indetinitely  small  in  every  direction, 
and  thougii  retaining  its  material  i)roperties  nuiy  be  treated 
as  a  geometric  point. 

Spdn'  is  indelinite  extension.  Time  is  any  limited  por- 
tion ol'  duration. 

5.  Rest  and  Motion.— A  body  is  at  rent  when  it  con- 
stantly  occupies    tiie  same   place  in  space.     A  body  is  in 


VELOCITY.  3 

iiiofion  when  the  body  or  its  part.s  occupy  successively  dif- 
lereiit  positions  in  space.  But  we  cuunot  judge  of  the  state 
of  rest  or  motion  of  a  body  without  referring  it  to  tlie 
jiositions  of  oti)er  bodies  ;  and  iience  rest  and  motion  must 
he  considered  as  necessarily  irlnfire. 

If  there  were  anvtliiiig  wliieii  we  knew  to  be  absolutely  lixed  in 
s])ace,  we  might  perceive  absolute  nuitiou  by  change  ct'  i)lace  with 
ret'erence  to  that  object.  But  as  we  know  of  no  such  thing  as  al so- 
lute rest,  it  follows  tliat  all  motion,  as  measured  by  us,  must  be 
relative  ;  i.  e.,  must  relate  to  sonu'thing  which  we  assume  to  be  tix<'(l 
Iience  the  same  thing  may  often  be  said  to  bo  at  rest  and  in  motion 
lit  the  same  time  ;  tor  it  may  Ix^  at  rest  in  regard  to  one  thing,  and  in 
motion  in  regard  to  another.  Fcr  exaini)le,  the  objects  on  a  vessel 
may  be  at  rest  with  reference  to  each  other  and  to  the  vessel,  while  they 
are  in  motion  with  reference  to  the  neighboring  shore.  So  a  man, 
punting  his  barge  up  the  river,  by  leaning  against  a  prJe  which  rests 
on  the  bottom,  and  walking  on  the  deck,  is  hi  nioti(m  relative  to  the 
barge,  and  in  motion,  but  in  a  different  manner,  relative  to  the  cur- 
vent,  while  he  is  at  rest  relative  to  the  earth. 

Mutiirii  ii^  nniforni  when  the  body  i)asses  over  equal  spaces 
in  e{iual  times  ;  otherwise  it  is  vai-iable. 

6.  Velocity. —  The  vcheilii  of  a  Imbj  is  Us  rate  of 
molion.  When  the  velocity  is  roii.s/dHf,  it  is  nieasuved  by 
Ihe  sjjace  ptissed  over  in  a  unit  of  time.  When  it  is  varin- 
ble,  it  is  measured,  at  any  instant,  by  the  space  over  which 
the  body  would  ptiss  in  a  unit  of  time,  Avere  it  to  move, 
during  that  unit,  with  the  same  velocity  that  it  has  at  the 
instttnt  considered. 

The  siHJcd  of  a  railway  train  is,  in  general,  variable.  If  we  were  to 
say,  for  exiinii)Ie,  that  it  was  running  at  the  rat(^  of  80  miles  an  hour, 
we  would  not  mean  that  it  ran  DO  miles  during  the  last  botir,  nor  that 
it  wouhl  run  30  miles  during  the  next  hour.     We  would  mean  that,  if 

it  were  to  run  for  a<i   hour  with   the  sp 1   which   it  tiow  has,  at  the 

instant  coiisid'red,  it  would  jiass  over  exactly  ItO  miles. 

In  order  to  have  ti  uniform  unit  of  velocity,  it  is  custom- 
ary to  c.\j)rcss  it  in  feet  and  seconds ;  and  when  velocities 


^M 


4  A  CCELERA  TIOX. 

arc  exiiresscd  in  any  otlicr  terms,  they  should  be  reduced  tn 
tlicir  equivalent  value  in  feet  and  seconds.  The  uini 
velocity,  therefore,  is  the  velocity  with  which  a  Ixulv 
describes  owc/wV  in  oiw  .second  ;  other  units  may  be  taken 
where  couveuieiiee  demands,  as  miles  and  hours,  etc. 

When  we  speak  of  the  space  passed  over  by  a  body,  we 
mean  the  p>t//i  or  line  which  a  point  in  the  body  or  which  a 
particle  describes. 

7.  Formulee  for  Velocity. — If  s  be  the  space  jiassed 
over  l)y  a  particle  in  /  units  of  time,  and  r  the  velocity,  it  is 
phiin  that,  for  uniform  velocif>/,  we  shall  have 


V  = 


V 


(1) 


that  is,  we  divide  the  whole  space  passed  over  by  the  time 
of  the  motion  over  tiuit  space. 

If  the  velocity  continually  changes,  equal  increments  are 
not  described  in  eijual  times,  and  the  velocity  beconu's 
a  function  of  the  time.  But  however  much  "the  velocity 
cluinges,  it  nuiy  be  regarded  as  constant  during  the 
intinitesimal  of  time  d/,  in  which  time  the  body  will 
descril)e  the  intinitesimal  of  space  ds.  Hence,  denoting  the 
velocity  at  any  instant  by  i\  we  have  * 

*  m 


V  = 


dt 


In  this  case  the  velocity  is  the  ratio  of  two  infinitesimals. 
These  two  expressions  for  the  velocity  are  true  whether  the 
]tarticle  be  moviiiji'  in  a  right,  or  in  a  curved,  line. 

8.  Acceleration  is  the  rale  nf  rlxoige  of  ralocitij.  Il 
is  a  velocity  iucrfineiit.  If  the  veU.eity  is  increasing,  the 
aceeh'ratioii  is  considered  positive ;  if  decreasing,  it  is 
negative. 

Acceleration    is   said   to  be   nniforni  when  the  velocity 


(2) 


MEAsvh-h:  OF  Ac<-i:i.KiiA'noy.  o 

receives  equal  iiuTeinent!:  in  ('(iiiul  tinie.s.     Otherwise  it  i-s 
variable. 

9.  Measure  of  Acceleration.— Uniform  acceleration 
is  measured  by  the  actual  increase  of  velocity  in  a  unit  of 
time.  Variable  acceleration  is  leasiiretl,  at  any  instant,  by 
the  velocity  which  would  be  generated  in  a  unit  of  time, 
were  the  velocity  to  increase,  during  that  unit,  at  the  same 
rate  as  at  the  instant  considered. 

Calling  /the  acceleration,  v  the  velocity,  and  t  the  time, 
wc  have,  when  the  acceleration  is  uniform. 


/  = 


t 


(1) 


However  variable  the  acceleration  is,  it  may  be  regarded 
as  constant  during  the  infinitesimal  of  time  dt,  in  which 
time  tlie  increment  of  velocity  will  be  dv.  Hence,  denoting 
the  accelerati...!  at  the  time  t  hy  f,  we  have 


/  = 


dv 
Jt' 


(2) 


We  also  have  (Art.  8) 
whioh  in  (2)  gives 


ds 


dv  d      ds 

J  ~  Tft  ~  dt  '  lit 


dfi 


(3) 


That  is,  when  the  acceleration  is  variable  it  is  measured,  at 
any  instant,  by  the  derivative  of  tiie  velocity  regarded  as  a 
function  of  the  time,  or  by  the  second  derivative  of  .he 
space  rej^arded  as  a  function  of  the  time. 

Ftom"(3)  we  get,  by  integration,  when /is  constant. 


/"  = 


d.< 
dt 


(4) 


ito^ 


6  VELOCITY   ASl)    ACrhLKUATIUN. 

and  2/s  =  «;', 

which  dotermine  the  velocity  and  space. 

10.  G-eometric  Representation  of  Velocity  and 
Acceleration. — The  velocity  of  ii  Iwdy  may  be  conveni- 
ently represented  geonietricully  in  niiig'iitiide  and  direction 
by  moans  of  a  straight  line.  Let  the  line  be  drawn  from 
the  point  at  which  the  motion  is  considered,  I'nd  in  the 
direction  of  motion  at  that  point.  With  a  convenient  scale, 
let  a  length  of  the  line  be  cut  off  that  shall  contain  as  many 
units  of  length  as  there  are  nnits  in  the  velocity  to  be  repre- 
sented. The  direction  of  this  line  will  represent  tiic 
direction  of  the  motion,  and  its  length  will  represent  the 
velocity. 

Also  an  acceleration  may  be  represented  geometrically  by 
a  straight  lino  drawn  in  the  direction  of  the  velocity 
generated,  and  ccntaining  as  many  units  of  length  as  there 
ai"e  units  of  acceleration  in  the  acceleration  considered. 
Also,  since  an  acceleration*  is  measured  by  the  actual 
increase  of  velocity  in  the  unit  of  time,  the  straight  line 
which  rei)reseiits  an  acceleration  in  magnitude  and  direc- 
tion will  also  completely  represent  the  velocity  generated  in 
the  unit  of  time  to  which  the  acceleration  corresponds. 

11.  The  Mass  of  a  body  or  particle  is  the  quantitij  of 
inal/rr  which  it  contains;  and  is  proportional  to  the 
Volume  and  Density  jointly.  The  Z>e«.'*)'^// may  therefore 
be  defined  as  the  quantity  of  matter  in  a  unit  of  volume. 

Let  M  be  the  mass,  p  the  density,  and  V  the  volume,  o' 
a  homogeneous  body.    Then  we  have 

M=  Vp,  (1) 

if  we  so  take  our  units  that  the  unit  of  mass  is  the  mass  ol 
the  unit  volume  of  a  bodv  of  unit  dcnsitv. 


If  tl 

uivc, 
lntc<rr 


♦  Uniform  acceleralion  U  here  meant. 


city  and 

B  eouvi'iii 
I  direction 
awn  from 
nd  in  tiie 
lient  scale, 
in  as  many 
o  be  roj)re- 
esent  the 
resent  tlio 

trically  by 
e  velocity 
h  as  tiiere 
'onsidcred. 
he  actual 
'aight  line 
and  diree- 
nerated  in 
londs. 

uantifij  of 
al    to   tlie 
jr  therefore 
volume, 
volnme.  o' 

(1) 
he  mass  ol 


MOMKSTVM.  i 

If  the  density  varies  from  point  to  point  of  tlie  body,  we 
have,  by  flic  above  formuhi,  and  the  notation  of  tlie 
Integral  Calculus, 

M  =  /ful  r  ^  ///fxlr  dif  <lz,  Ci) 

where  f)  is  supposed  to  be  a  known  function  of  .-r,  //,  z. 

In  England  the  unit  of  mass  is  th.e  imperial  standard 
pound  avoirdupois,  which  is  the  mass  of  a  certain  piece  of 
platinum  preserved  at  the  standard  ottice  in  London.  On 
the  continent  of  Europe  the  unit  of  mass  is  the  gramme. 
This  is  known  as  the  nbmlulc  or  kinetic  unit  of  mass. 

12.  The  Quantity  of  Motion,*  or  the  Momentum 

of  a  body  moving  without  ro'.ation  is  the  product  of  its 
mass  and  velocity.  A  double  mass,  or  a  double  velocity, 
would  correspond  to  a  double  quantity  of  motion,  and 
so  on. 

Hence,  if  we  take  as  the  unit  of  momentum  the  mo- 
mentum of  the  unit  of  mass  moving  with  the  unit  of 
velocity,  the  momentum  of  a  mass  M  moving  with  velocity 
V  is  Mv. 

13.  Change  of  Quantity  of  Motion,  or  Change  of 
Momentum,  is  proportional  to  the  mass  moving  and  the 
change  of  its  velot.iy  jointly.  If  then  the  mass  remains 
constant  the  change  of  momentum  is  measured  by  the 
product  of  the  mass  into  the  change  of  velocity  ;  and  the 
mte  of  chnnge  of  momentum ,  or  acceleration  of  momentnm, 
is  measured  by  the  product  of  the  mass  maving  and  the 
rate  of  ch.ange  of  velocity,  that  is,  by  the  product  of  the 
mass  moving  and  the  acceleration  (Art.  8).  Thus,  calling 
M  the  mass,  we  have  for  the  measure  of  tlie  rate  of  change 
of  momentum, 


M 


d^ 
dt^' 


*  This  phraee  was  used  by  Newton  in  place  of  the  more  modern  term  "  Momen- 
tum." 


^M 


8 


STAT/c  .)//;.i.s77i'a;   or  force. 


14.  Force. — Force  is  diiy  cause  which  changes,  or  teyuh 
to  chfuifje,  (I  body's  stitle  of  rest  or  motion. 

X force  alway.s  tends  to  jirodiici'  niotidii,  but  may  be  pn- 
voiited  froii!  actually  pnitliiciii^'  it  by  tlio  coimttTactioii  df 
ati  iMiiial  and  (ippositc  t'orcc.  Sovoral  forirs  may  so  nci  un 
a  ImkU  as  to  ncntralizo  caeb  otlicr.  Wlien  a  body  ivniain.- 
at  rest,  thon^jb  aited  on  by  forces,  it  is  said  to  be  in 
erjuilibriiun;  or.  in  otber  words,  tlie  forces  are  said  to 
produce  e(|uilibrinm. 

Wliat  force  is.  in  its  nature,  we  do  not  know.  Forces 
are  known  to  us  only  by  ibcir  effects.  In  order  to  measure 
them  we  must  com|iare  the  etfects  which  tliey  produce 
under  the  same  cin'umstances. 

15.  Static  Measure  of  Force.— 77<e  effect  of  a  force 
depends  on:  1st,  its  niiii/ni/iittc,  or  intensity ;  2d,  its  direc- 
tion; i.e.,  the  direction  in  whicli  it  tends  to  move  tiie  body 
on  wliich  it  acts  :  and  :]t\.  its  /miiif  o/ap/dicntion  ;  i.e.,  the 
point  at  wliicb  tiie  force  is  applied. 

The  effect  of  a  force  is  pressure,  and  may  be  expressed  bv 
the  weiu:ht  which  will  counteract  it.  Every  force,  statically 
considered,  is  a  pressure,  and  hence  has  magnitude,  and 
may  l)e  measured.  A  force  may  i)roduce  motion  or  not. 
according  as  the  body  on  which  it  acts  is  or  is  not  free  to 
move.  For  example,  take  the  case  of  a  body  resting  on  a 
table.  'J"he  same  force  which  jiroduces  ]iressure  on  the 
table  would  cause  the  body  to  fail  toward  the  earth  if  the 
table  were  removed. 

The  cause  of  this  pressure  or  motion  is  gravity,  or  the 
force  of  attraction  in  the  earth.  In  the  lirst  case  the  attrac- 
tion f  the  earth  produces  a  pressure;  in  the  second  case  it 
produces  motion.  Now  either  of  these,  viz.,  the  pressure 
which  the  body  exerts  wlien  at  rest,  or  the  (piantity  of 
motion  it  accjuires  in  a  unit  of  time,  may  be  taken  as  a 
means  of  measuring  the  magnitude  of  the  force  of  attrac- 
tion that  tlie  earth  exerts  ou  the  body.     The  former  i-: 


nges,  or  tetuU 

i  may  bo  \\tv- 
ntfractiim  of 
may  so  aci  nn 
Ixuly  ivi!iaiii> 
said  to  be  in 
^  are   said  to 

now.  Forces 
T  to  measure 
tliey  produce 

ed  of  a  force 
2d,  its  (linr- 
110 ve  the  body 
ion  ;  i.e.,  the 

'  expressed  by 
rce.  statically 
ignitude,  and 
otion  or  not. 
is  not  free  to 
resting  on  a 
ssuro  on  the 
e  earth  if  the 

:ravity.  or  the 
ise  the  attrac- 
:eeond  case  it 
the  pressure 
L'  <|uantity  of 
be  taken  as  a 
)rce  of  attrac- 
'hc  former  is 


METirOI)    OP    COMPARING    FORCES. 


d 


called  the  sfafir  method,  and  the  forces  are  called  static 
forces;  the  latter  is  called  the  kinetic  method,  and  the 
forces  are  called  kiiirtic  forces.  Wri(//it  is  the  name  givi'ii 
to  the  pressure  which  the  attraction  of  the  earth  causes  a 
body  to  exert.  Hence,  since  static  forci's  produce  pressure, 
we  may  take,  as  the  unit  of  force,  a  pre.ssiire  of  ow  poini'l 
(Art.  11). 

Therefore,  t/ie  mae/nifudc  of  a  force  may  he  measured 
statieallij  hy  the  pres.wre  it  will  produce  upon  some  hody, 
and  e.rpressed  in  pounds.  This  is  called  the  Static  measure 
of  force,  and  its  unit,  one  pound,  is  called  the  Gravitation 
unit  of  force. 

16.  Action  and  Reaction  are  always  equal  and 
opposite. — This  is  a  law  of  nature,  and  our  knowledge  of 
it  comes  from  experience.  If  a  force  act  on  a  body  hold  by 
a  fixed  obstacle,  the  latter  will  oi)pose  an  eciual  and  con- 
trary resistance.  If  the  force  act  on  a  body  free  to  move, 
motion  will  ensue  ;  and,  in  the  act  of  raoYing,  the  inertia 
of  the  body  will  oppose  an  equal  and  contrary  resistance. 
If  we  press  a  stone  with  the  hand,  the  stone  presses  the 
hand  in  return.  If  we  strike  it,  we  receive  a  blow  by  the 
act  of  giving  one.  If  we  urge  it  so  as  to  give  it  motion,  we 
lose  some  of  the  motion  which  we  should  give  to  our  limbs 
by  the  same  effort,  if  the  stone  did  not  impede  them.  In 
each  of  these  cases  there  is  a  reaction  of  the  same  kind  as 
the  action,  and  equal  to  it. 

17.  Method  of  Comparing  Forces.— Two  forces  arc 
equal  when  being  applied  in  opposite  directions  to  a 
particle  they  maintain  equilibrium.  If  we  take  two  equal 
forces,  and  apply  them  to  a  particle  in  the  same  direction, 
we  obtain  a  force  double  of  cither  ;  if  we  unite  three  equal 
forces  we  obtain  a  triple  force  ;  and  so  on.  So  that,  in 
general,  to  compare  or  measure  forces,  we  have  only  to 
adopt  the  same  method  as  who  i  we  compare  or  measure 


^ 


10 


REPRESENTATION    OF   FORCES. 


any  (luantitios  of  the  Siimo  kind  ;  tliat  is,  we  must  take 
some  known  force  as  tiie  tinil  offoi'a\  and  tlivjii  express,  in 
ninnhers,  tlie  rehiMon  which  the  otiier  forces  bear  to  th it 
nieasurinj^  unit.  For  example,  if  one  pound  i)c  tlie  unit  of 
force  (Art.  15),  a  force  of  12  pounds  is  expressed  by  12; 
and  so  on. 

18.  Representation  of  Forces  by  Symbols  and 
Lines. — If  1*.  Q.  K.,  etc.,  represent  forces,  they  are  number.'* 
expressing  tlie  number  of  times  which  the  concrete  unit  of 
force  is  contained  in  tiie  given  forces. 

Forces  may  be  rej)resented  <ri'ometrically  by  right  lines; 
and  tliis  mode  of  representation  iias  the  advantage  of  giving 
the  direction,  magnituue,  and  point  of  appUcation  of  each 
force.  Tims,  draw  a  line  in  the  direction  of  the  given 
force ;  then,  having  selected  a  unit  of  lengtli,  such  as  an 
inch,  a  foot,  etc.,  measure  on  tiiis  hue  as  many  units  of 
length  as  the  given  force  contains  units  of  weight.  The 
viagnifiide  of  the  force  is  rei)resented  by  the  measured 
length  of  the  line ;  Mi  direction  by  the  direction  in  which 
the  line  is  drawn;  and  \i&  point  of  application  by  the  point 
from  which  the  line  is  drawn.* 

Thus,  let  the  force  P  act  at  the  point    * ? 

A,   in    the    direction   AB,   and    let    AB  '^'9-  '• 

represent  as  many  units  of  length  as  P  contains  units  of 
force;  then  the  force  P  is  represented  geometrically  l)y 
the  line  AB;  for  the  force  acts  in  the  direction  from  A 
to  B;  its  point  of  application  is  at  A,  and  its  magnitude  is 
represented  by  the  length  of  the  line  AB. 

19.  Measure  of  Acceleratiug  Forces.— From  our 
definition  ot  force  (Art.  14),  it  is  clear  that,  when  a  single 

•  Forcee,  vulocitloi*,  nnd  iiccelcratlont!  arc  lUrecleil  i/tian/l/lm,  nii<l  ho  niny  ln' 
repn>i<eiU(Hl  by  a  liiiu,  in  (lircctiun  and  inai;nitiidt',  uiid  may  be  conipouiuled  in  t 
game  way  as  vectors. 

If  anything  ha?  maRnitiulo  and  ciiruction,  th"  magnitude  and  dirucUun  lakvii 
togelhur  cuDBtitutn  a  rector. 


B  must  take 
ti  oxprt'ss,  ill 
bear  to  t,lii>; 
!  tlio  unit  of 
ssecl  by  12; 

nbols   and 

ire  imrril)ors 
rete  unit  of 

right  lines; 
ge  of  giving 
iiou  of  each 
f  the  given 
Huch  as  an 
iiy  units  of 
)ighfc.  The 
J  measured 
1  in  whicli 
y  tlic  point 


Fig.  I. 

ins  units  of 
trically  l)y 
m  from  A 
ignitiulo  is 

From  our 
11  a  single 

111(1  HO  may  lio 
ouiuled  in       ) 

llructtun  tukeii 


MEASVnB    OF   ACCELBttAflNG    FOliCKS. 


11 


force  acts  upon  a  particle,  perfectly  free  to  move,  it  must 
produce  motion  ;  and  hence  the  force  may  he  represenleii 
to  us  by  (he  motion  it  has  produced.  Mut  motion  is 
meiisured  in  terms  of  velocity  (Art.  0),  and  conse(iuently  the 
velocity  communicated  to,  or  impressed  upon,  a  particle,  in 
a  given  time,  may  be  taken  as  a  measure  of  the  force. 
That  is,  if  tiie  same  particle  moves  along  a  right  line  so 
that  its  velocity  is  increased  at  a  constant  rate,  it  will  be 
acted  upon  by  a  constant  force.  If  a  certain  constant  force, 
acting  for  a  second  on  a  given  particle,  generate  a  velocity 
of  32.2  feet  per  second,  a  double  force,  acting  for  one 
second  on  the  same  particle,  woiUd  generate  a  velocity  of 
64.4  feet  per  second  ;  a  triple  force  would  generate  a 
velocity  of  DO.O  feet  per  second,  and  so  on. 

if  the  rate  of  increase  of  the  velocity,  (/.  e.,  the  accelera- 
tion), of  the  particle  is  not  uniform,  the  force  acting  oi\  it 
is  not  uniform,  and  the  magnitude  of  the  force,  at  any 
point  of  the  particle's  path,  is  measured  by  the  acceleration 
of  the  particle  at  this  point.  Hence,  since  one  and  the 
same  particle  is  capable  of  moving  with  all  possible  accelera- 
tions, all  forces  may  he  measured  by  the  velocities  they 
generate  in  the  same  or  erjual  particles  in  the  same  or  equal 
times.  When  forces  are  so  measured  they  are  called 
A  cede  rat  in;/  Forces. 

20.  Kinetic  or  Absolute  Mea£  ire  of  Force.*— Let 
n  equal  particles  be  i)laced  side  by  siile,  and  let  each  of  them 
be  acted  on  uniformly  for  the  same  time,  by  the  same  force. 
Each  i)article,  at  the  end  of  this  time,  will  have  the  same 
velocity.  Now  if  these  n  separate  jmrticles  are  all  united  so 
as  to  form  a  body  of  n  times  the  mass  of  each  particle,  and 
if  each  one  of  them  is  still  acted  on  by  the  same  force  as 

•  Arts.  80,  91,  88,  and  98,  trout  of  the  Kindle  incusiirc  of  force,  und  may  be 
omitted  till  Pnrt  III  la  rcnclied  ;  but  it  Ih  coiivcnli^iit  to  |iiewnt  tliem  once  for  all, 
aivl,  for  the  pake  of  reference  mnl  conipiuiHoii,  to  place  them  with  the  Htatic 
meaxuro  of  force  lit  th(^  beginning  of  the  work. 


ita 


IZ       KLXETir    OR    AliSOLlTTK    3tEASVRE    OF   FOncK. 

before,  tliis  body,  at  the  end  of  the  time  considered,  will 
iiave  tiu-  same  velocity  that  each  separate  partiei.?  Iiad,  and 
will  be  acte<l  on  l>y  n  limes  (be  force  which  jijeneraled  this 
velocity  in  tiie  particle.  Comparing);  a  sin<rle  particle,  then, 
with  the  body  whose  mass  is  n  times  the  mass  of  tiiis 
particle,  v.  v'  see  thar,  to  produce  the  same  velocity  in  two 
bodies  by  forces  acting  on  them  for  the  same  time,  the 
magnitudes  of  the  forces  must  be  proportional  to  the 
masses  on  which  they  act.*  Ilenee,  generally,  since  force 
varies  as  the  velocity  when  the  mass  is  constant  (Art.  ID), 
and  varies  as  the  nuiss  when  the  velocity  is  constant,  we 
have,  by  the  ordinary  law  of  proportion,  when  both  are 
changed,  force  varies  as  the  product  of  the  mass  acted  ujjon 
and  the  velocity  generated  in  a  given  time  ;  that  is,  it  varies 
as  the  qnantity  of  moti  w  (Art.  13)  it  produces  in  a  given 
mass  in  a  given  time.  If  the  force  be  variable,  the  rate  of 
change  of  velocity  is  variable  (Art.  19),  and  hence  the  force 
varies  as  the  ))roduct  of  the  mass  on  which  it  acts  and  the 
rrt/e  o/'r//rt«//t' of  velocity,  i.e.,  it  varies  as  the  <ureh'r(ifion 
of  the  momentum  (Art.  l!]).  Therefore,  if  any  force  P  act 
on  a  mass  M,  we  have 


P^  Mf', 
or,  in  the  form  of  an  equation 

/'  =  k-Mf, 


(1) 


(2) 


where  k  is  some  constant. 

If  the  unit  of  force  be  taken  as  that  force  which,  acting 
on  the  unit  of  mass  for  the  unit  of  time,  generates  the  unit 
of  velocity,  then  if  we  put  M  equal  to  unity,  /'.  c,  take  the 
unit  of  mass,  and  /'equal  to  unity,  i.  e.,  take  the  unit  of 
acceleration,  we  must  have  the  force  producing  the  accel- 
eration ecpial  to  the  unit  of  force,  or  P  equal  to  unity, 


♦  MliiclilnV  Sialics  p,  D. 


rfMM*i 


■Hi 


tm 


liilorod,  will 
'■■■■}.  liacl,  and 
iioralcd  tiiis 
rticlo,  tliiMi. 
ass  of  fhis 
icity  in  two 
i  time,  the 
nal  to  the 
sir.oe  force 
it  (Art.  19), 
t>iKstant,  we 
II  botli  are 
acted  upon 
is,  it  varies 
in  a  given 
the  rate  of 
ce  tlie  force 
ts  and  the 
•iccelerdlion 
oi'cc  P  act 


(1) 


(2) 


lich,  acting 
;e8  tlie  unit 
;.,  take  the 
the  unit  of 
;  the  accei- 
l  to  unity. 


THE  AliSOhfTE  on   Kr\KTlC  MEASl'HE   OF  FORCE.    13 

Heuce  k  must  also  be  eciual  to  unity,  and  we  iiavc  the 
equation, 

P  =  Mf.  (3) 

Therefore,  tlic  Kinetic  or  Abmlnle  measure  of  a  force  is 
llw  rale  of  chanije  or  acceleration*  (f  momentum  it  produces 
ill  a  uiiit  of  time. 

If  the  force  is  constant,  (3)  becomes  by  (1)  of  Art.  9, 


P  = 


Mv 


(*) 


And  if  the  force  is  variable,  (3)  becomes  by  (3)  of  Art.  9, 

(5) 


21.  The  Absolute   or  Kinetic  Unit  of  Force.— 

A  second,  a  foot,  and  a  pound  being  tlie  units  of  time,  space, 
and  ma.ss,  respectively  (Arts.  6  and  11),  we  are  reciuired  to 
tind  tiie  corresponding  unit  of  force  that  the  above  ecjuation 
nuiy  be  true.  The  unit  of  force  is  that  force  which,  actimj 
for  one  second,  on  ttie  muss  of  one  jtound,  yeneratcs  in  it  a 
velocity  of  one  foot  per  second.  Now,  from  the  results  of 
numerous  experiments,  it  has  been  ascerrained  that  if  a 
body,  wcigliing  one  pound,  fall  freely  for  one  second  at  the 
sea  level,  it  will  acquire  a  velocity  of  about  M.i  feet  per 
second ;  i.  e.,  a  force  ecjual  to  the  weight  of  a  pound,  if 
acting  on  the  mass  of  a  pound,  at  the  sea  level,  generates  in 
it  in  one  second,  if  iVee  to  move,  a  velocity  of  nearly  ^'Vi 
feet  per  second.      It   follows,  therefore,  that  a  force  of 

,- ^    of  the  weight  of  a  pound,   if  acting  on  the  mass  of 

M  pound,  at  the  sea  level,  generates  in  it  in  one  second,  if 
free  to  move,  a  velocity  of  one  foot  per  second  ;  and  hence 

♦  Bw  T»U  aud  8toi'l«'«  UyuBiulvH  of  i  Pnrllclo.  p.  43. 


^ 


14 


ilEASl'RES   OF  FORCE. 


the  unit  of  force  is  j^^^  of  the  weight  of  a  pound,  or  rather 

less  than  the  weight  of  iialf  an  ounce  avoirdupois  ;  so  tiiat 
lialf  an  ounee,  acting  on  the  mass  of  a  pound  for  one 
second,  will  give  to  it  a  velocity  of  one  foot  i)er  second. 
This  is  the  British  abmlutc  kineiic*  unit  of  force. 

In  order  tiiat  Eq.  3  (Art.  ^0)  may  be  uni\ersally  true 
when  a  second,  a  foot,  and  a  pound  are  the  units  of  time, 
space,  and  nuiss  respectively,  all  forces  must  be  expressed  in 
terms  of  this  unit. 

22.  Three  Ways  of  Measuring  Force.— (1.)  If  a 

force  does  not  produce  motion  it  is  measured  by  the  pres- 
sure it  produces,  or  the  number  of  pounds  it  will  sujiport 
(Art.  15).  This  is  tiie  measure  of  Static  Force,  and  its 
unit  is  the  weiyhf  of  a  poutnl. 

{'i.)  If  we  consider  forces  as  always  acting  on  a  unit  of 
muss,  and  suppose  that  there  are  no  forces  acting  in  the 
opposite  direction,  then  these  forces  will  .be  measured 
siini)ly  by  the  velocities  or  accelerations  which  tiiey  generate 
in  a  given  time.  This  is  the  measure  of  Accelerating  Force, 
and  its  unit  is  that  force  which,  acting  on  the  unit  of  mass, 
during  the  unit  of  time,  generate  the  unit  of  velocity; 
hence  (Art.  21),  the  unit  of  force  is  the  force  tvhich,  acting 
on  one  pound  of  mass  for  one  second,  generates  a  velocity  of 
one  foot  per  second. 

(3.)  If  forces  act  oti  different  masses,  and  produce  motion 
in  them,  and  we  consider  as  before  that  there  are  no  f(u-ccs 
acting  in  the  opposite  directioTi,  then  the  forces  are  meas- 
ured tig  the  quantity  of  motion,  or  by  the  acceleration  of 
momentum  generated  in  a  unit  of  time  (Art.  20).  This  is 
the  measure  of  Muring  Force,  and  its  nnil  (Art.  21)  is  the 
force  vhichy  (ictiiig  on  one  pound  if  mass  for  one  second, 
tfueratcs  a  velocity  of  one  foot  per  second. 


*  (ntriHlucod  by  GausB, 


MEANING    OF  O   IN  DYNAMICS. 


16 


^,  or  rather 

•is  ;  so  tliiit 
lul  for  Olio 
per  sceoiul. 

'0. 

crsally  trut 
its  of  time, 
xpressed  in 


-(1.)  If  a 
)y  the  pres- 
k'ill  .siijiport 
'cc,  and  its 

a  unit  of 
ting  in  the 

measured 
cy  generate 
'ting  Force., 
it  of  viass, 
f  veloeily; 
lich,  acting 
velocity  of 

Hce  motion 
B  no  forces 

are  meas- 
'erafion  of 
L     This  'is 

*^1)  is  the 
inv  second, 


It  must  be  understood  that  when  we  speak  of  static, 
accelerating,  or  moving  forces,  we  do  not  refer  to  different 
kinds  of  force,  but  only  to  force  as  measured  in  different 
ways. 

23.  Meaning  of  f/  in  Dynamics. — The  most  impor- 
tant case  of  a  consiaut,  or  very  nearly  constant,  force  is 
gravity  at  the  surface  of  the  earth.  The  force  of  gravity  u 
so  nearly  constant  for  places  near  the  earth's  surface,  that 
falling  bodies  may  be  taken  as  examples  of  motion  under  a 
constant  force.  A  stone,  let  fall  from  rest,  moves  at  first 
very  slowly.  During  the  first  tenth  of  a  second  the  velocity 
is  very  small.  In  one  second  the  stone  has  acquired  a 
velocity  of  about  32  feet  per  second. 

A  great  number  of  experiments  have  been  made  to  ascer- 
tain tiie  exact  velocity  which  a  body  would  acquire  in  one 
second  under  the  action  of  gravity,  and  freed  from  the 
resistance  of  the  air.  The  most  accurate  method  is  indi- 
rect, by  means  of  the  pendulum.  The  result  of  pendulum 
experiments  made  at  Leith  Fort,  by  Captain  Kater,  is, 
that  the  velocity  acquired  by  a  body  falling  unresisted  for 
one  second  is,  at  that  place,  33.207  feet  per  second.  The 
velocity  acquired  in  one  second,  or  the  acceleration  (Art. 
8),  of  a  body  falling  freely  in  vacuo,  is  found  to  vary 
slightly  with  tiio  latitude,  and  also  with  the  elevation  above 
the  sea  level.  In  London  it  is  32.1889  feet  per  second.  In 
latitude  45°,  near  Hordeaux,  it  is  32.1703  feet  per  second. 

This  acceleration  is  usually  denoted  by  r/ ;  and  when  wo 
say  that  at  any  place  g  is  equal  to  32,  we  mean  that  the 
velocity  generated  i)or  second  in  a  body  falling  freely* 
under  the  action  of  gravity  at  that  place,  is  a  velocity  of 
32  feet  per  sfcond.  The  average  value  of  g  for  the  whole 
of  Great  Britain  differs  but  little  from  32.2  ;  and  hence  the' 
nnincrical  value  of//  for  that  country  is  taken  to  be  32.2. 

♦  A  l)o<ly  i»  Huiil  to  l)c  moviug//*Wy  wlicu  It  U  acted  upon  by  no  forcc8  except 
tlioi*e  imiler  voueiilcratlou, 


^^ 


16 


TJUH   UNIT  OF  MASS. 


The  formulii,  deduced  from  observation,  and  a  certain 
theory  regarding  the  figure  and  density  of  the  earth,  which 
may  be  employed  to  calculate  the  most  probable  value  of 
the  apparent  force  of  gravity,  is 

g  z=z  0{\  ■\-  .005133  sin^  A), 

where  O  is  the  apparent  force  of  gravity  on  a  unit  mass  at 
the  equator,  and  g  the  force  of  gravity  in  any  latitude  A; 
the  value  of  O,  in  terms  of  the  British  absolute  unit,  being 
32.088.     (See  Thomson  and  Tait,  p.  226.) 

24.  Gravitation  Units  of  Force  and  Mass.— If  in 

(3)  of  Art.  20,  we  put  for  P,  the  weight  W  of  the  body, 
and  write  g  for  /  since  we  know  the  acceleration  is  </,  (3) 
becomes 

W  =  mg.  (1) 


W 

m  =  — • 

g 


(2) 


w 


and  hence  —  may  be  taken  as  the  measure  of  the  mass. 


In  gravitation  measure  forces  are  measured  by  the  pres- 
sure they  will  produce,  and  the  unit  of  force  is  one  pound 
(Art.  15),  and  the  unit  of  mass  is  the  quantify  of  matter  in 
a  body  winch  weighs  (j  pounds  at  that  place  where  the  accel- 
eration of  gravity  is  g. 

This  definition  gives  a  unit  of  mass  which  is  constant  at 
the  same  place,  but  changes  with  the  locality ;  i.  e.,  its  weight 
clianges  with  the  locality  while  the  quantity  of  matter  in  it 
remains  the  same.  Thus,  the  unit  of  mass  would  weigh  at 
Hordeaux  32.K03  pounds  (Art.  23),  while  at  Leith  Fort  it 
would  weigh  32.207  pounds.  Ijct  m  be  the  mass  of  a  l)o(iy 
which  weighs  w  pounds.  The  (|uantity  of  matter  in  this 
hody  remains  the  same  when  carried  from  i)lace  (o  placi'. 
If  it  were  possible  to  transport  it  to  another  planet  its  mass 


GRAVITATIOX  MEASURE  OF  FORCE. 


17 


id  a  certain 
arth,  which 
ble  value  of 


init  mass  at 
latitude  A; 
unit,  being 


ass.— If  in 

f  the  body, 
iou  is  g,  (;3) 


(1) 
(2) 


10  mass. 


by  the  pres- 
i  one  jwiind 
\f  matter  in 
'0  the  accel- 


constiuit  at 
p.,  its  weight 
matter  in  it 
Id  weigh  at 
litii  Fort  it 
i  of  a  l)o(ly 
Iter  in  this 
'e  to  |>hur. 
lot  its  mass 


would  not  be  altered,  but  its  weight  would  be  very  different. 
Its  Wi'ight  wherever  placed  would  vary  directly  as  the  force 
of  gravity  ;  but  the  acceleration  also  would  vary  directly  as 
the  force  of  griivity.  If  placed  on  the  sun,  for  example,  it 
would  weigh  about  ^8  times  as  much  as  on  the  surface  of 
the  earth  ;  but  the  acceleration  on  the  sun  would  also  be 
i8  times  as  much  as  on  the  surface  of  the  earth  ;  that  is, 
the  ratio  of  the  weight  to  the  acceleration,  anywhere  in 

ir 

the  universe   is  constant,   and   hence    — ,  which    is  the 

U 
numerical  value  of  m  (Eq.  3),  is  constant  for  the  same 

mass  at  all  places. 

25.  Comparison  of  Gravitation  and  Absolute 
Measure. — The  pound  weight  has  been  long  used  for  the 
measuremeut  of  force  instead  of  mass,  and  is  the  recognizeil 
standard  of  reference.  It  came  into  general  use  because  it 
afforded  the  most  ready  and  simple  method  of  estimating 
forces.  Tiie  pressure  of  steam  in  a  boiler  is  always  reck- 
oned in  jmunds  per  square  inch.  The  tension  of  a  string  is 
estimated  in  pounds;  the  force  necessary  to  draw  a  train  of 
cars,  or  the  pressure  of  water  against  a  lock-gate,  is 
expressed  in  pounds.  Such  e.\i)re8sions  as  "a  force  of 
10  poui\ds,"  or  "a  pressure  of  steam  equal  to  50  pounds  on 
the  inch,"  are  of  every  day  occurrence.  Therefore  this 
method  of  measuring  forces  is  eminently  convenient  in 
practice.  For  this  reason,  and  because  it  is  the  one  used 
by  most  engineers  and  writers  of  median ii's,  we  shall  adopt 
it  in  this  work,  and  adhere  to  the  measurement  of  force  by 
pounds,  and  give  all  our  results  in  the  usual  gravitation 
measure.     In  this  measure  it  is  convenient  to  represent  the 

W 

mass  of  a  body   weighing   IT  pounds  by  the  fraction  — 

(.Vrt.  24),  80  (hat  (3)  of  Art.  30  becomes 


(1) 


^ 


18  EXAMPLES. 

To  do  so  it  will  only  be  necessary  to  assume  that  the  unit 
of  mass  is  the  iiiiatitity  of  matter  in  a  body  weighing  y 
pounds,  and  changes  in  weight  in  the  same  proportion  that 
g  changes  (Art.  24). 

Of  course,  the  units  of  mass  and  force  in  (3)  of  Art.  20 
may  be  either  absolute  or  gravitation  units.  If  absolute. 
the  unit  of  mass  is  one  pound  (Art.  11),  and  the  unit  of 

force  is  -  pounds  (Art.  21).     If  gravitation,  the  units  are 

(J  times  as  great;  i.  e.,  the  unit  of  mass  is  (/  pounds  (Art. 
24),  and  the  unit  of  force  is  one  pound  (Art.  15). 

The  advantage  of  the  gravitation  measure  is,  it  enables  us 
to  express  the  force  in  pomids,  and  furnishes  us  with  a  con- 
stant numerical  representative  for  the  same  quantity  of 
matter ;  that  is  to  say,  a  mass  represented  by  20  on  tiie 
equator  would  be  represented  by  20,  at  the  pole  or  on 
the  sun.  Hence,  in  (1),  P  is  the  static  measure  of 
any  moving  force  [Art.  23,  (3)],  W  is  the  tveight  of  the 
body  in    pounds,  (j  the  acceleration  of  gravity  (Art.  23), 

W 

—  the  mass  upon  which  the  force  acts  [(3)  of  Art.  24],  and 

which  is  free  to  move  under  the  action  of  F,  the  unit  of 
mass  being  the  mass  weighing  g  pounds,  and  /  the 
acceleration  which  the  force  P  produces  in  the  mass. 

EXAMPLES 

1.  Compare  the  velocities  of  two  points  which  move 
uniformly,  one  through  5  feet  in  half  a  second,  and  the 
other  through  100  y&,rd8  in  a  minute.    Ans.  As  2  is  to  1. 

3.  Compare  the  velocities  of  two  points  which  move  uni- 
formly, one  through  720  feet  in  one  minute,  and  the  other 
thro  igli  34  yards  in  three-quarters  of  n,  second. 

Am.  As  0  is  to  7. 

3.  A  railway  train  travels  100  miles  in  2  hours  ,  'inil 
the  average  velocity  in  feet  per  second.  Ans.  73^. 


EXAMPLES. 


19 


hat  the  unit 

weighing  y 

portion  that 

)  of  Art.  20 

If  absoluk\ 

.  the  unit  of 

le  units  are 

(ounds  (Art. 

it  enables  us 
I  with  a  con- 
quantity  of 
•  20  on  the 
pole  or  on 
measure  of 
ciffJit  of  tlio 

(  (Art.  a;j), 

irt.  24],  and 

the  unit  of 
and  /  the 
mass. 


vhich  niove 
ul,  and  the 
2  is  to  1. 

Ii  move  uni- 
d  the  other 

C  is  to  7. 
Iiours  ,   *in(l 
ins.  73f 


'  4.  One  point  moves  uniformly  round  the  circumferenoe 
of  a  circle,  while  another  point  moves  uniformly  along 
the  diameter  ;  compare  their  velocities. 

Ans.  As  n  is  to  1. 

■  5.  Supposing  the  earth  to  be  a  sphere  25000  miles  in 
circumference,  and  turning  round  once  in  a  day,  deter- 
mine the  velocity  of  a  point  at  the  equator.     • 

Ans.  1527^  ft.  per  sec. 

6.  A  body  has  described  50  feet  from  rest  in  2  seconds, 
with  uniform  acceleration  ;  find  the  velocity  acquired. 

From  (5)  of  Art.  9  we  have 

/=25; 

and  from  (4)  we  have      ft  =  v; 

.'.    V  =  50. 

I 

^^.  Find  the  time  it  will  take  the  body  in  the  last  exam- 
ple to  move  over  the  next  150  feet. 

From  (5)  of  Art.  9  we  have 

a  =  ift*;    .-.    etc 

Ans.  2  seconds. 

8.  A  body,  moving  with  uniform  acceleration,  describes 
63  feet  in  the  fourth  second  ;  find  the  acceleration. 

Ans.  18. 

9.  A  body,  with  uniform  acceleration,  describes  72  feet 
while  its  velocity  increases  from  IG  to  20  feet  per  second  ; 
find  the  whole  time  of  motion,  and  the  acceleration. 

Ans.  20  seconds ;  1. 

^10.  A  body,  in  passing  over  9  feet  with  uniform  accelera- 
tion, iuis  its  velocity  increased  f^m  4  to  5  feet  per  second ; 
lind  the  whole  space  described  from  rest,  and  the  accelera- 
fi,)„_  Ans.  25  feet ;  4. 


I* 


20 


aXAMl'LES. 


11.  A  body,  uniformly  accelerated,  is  found  to  l)e  mov- 
ing ut  the  end  of  10  .seconds  with  a  velocity  wiiich  if 
continued  uniformly,  would  carry  it  through  45  miles'  in 
the  next  hour  ;  tind  the  acceleration.  Ans.  ()|. 

12.  Find  the  mass  of  a  straight  wire  or  rod,  the  density 
of  which  varies  directly  as  the  distance  from  one  end. 

Take  the  end  of  the  rod  as  origin  ;  let  a  =  its  length  • 
let  the  distance  of  any  point  of  it  from  that  end  =  x  ■ 
and  let  w  =  the  area  of  its  transverse  section,  and  k  =  the 
density  at  the  unit's  distance  from  the  origin.     Then 

dV=u(lx;    and    p  =  kx; 
and  (2)  of  Art,  11  becomes 

13.  Find  the  mass  of  a  circular  plate  of  uniform  thick- 
ness, the  density  of  which  varies  as  the  disUince  from  the 
centre. 

Ans.  ^TTMa^  where  a  is  the  radius,  k  the  density  at 
the  unit's  distance,  and  //  the  thickness. 

'14.  Find  the  mass  of  a  sphere,   whose  density   varirs 
inversely  as  the  distance  from  the  centre. 
J»w.  2nm%  where  p  is  the  density  of  the  outside  stratum. 


I  to  be  niov- 
ty  wliich,  if 
45  miles  in 
Anti.  (J|. 

I,  the  density 
le  end. 

:  its  lengtii  ; 
it  end  =  X  ; 
ind  i-  =  the 
Then 


STATICS     (REST). 


iform  thick- 
ce  from  the 

>  density  at 

1  si ty   varies 
idc  stratum. 


CHAPTER     II. 

THE    COMPOSITION     AND     RESOLUTION     OF    CONCUR- 
RING     FORCES— CONDITIONS    OF    EQUILIBRIUM. 

26,  Problem  of  Statics.  —The  primary  conception  of 
force  is  tiiat  of  a  cause  oi"  motion  (Art.  14).  If  only  one 
force  acts  on  a  particle  it  is  clear  tliat  the  particle  cannot 
remain  at  rest.  In  statics  it  is  only  tlie  tcndrnci/  which 
forces  have  to  produce  motion  that  is  considered.  IMierc 
must  be  at  least  two  forces  in  statics ;  and  they  are  con- 
sidered as  acting  so  as  to  counteract  each  otiier's  tendency 
to  cause  motioti,  thereby  producing  a  state  of  equilibrium 
in  the  bodies  to  which  they  arc  applied.  The  forces  which 
act  upon  a  body  may  be  in  e(iuilibrium,  and  yet  motion 
exist;  but  in  such  cases  the  motion  is  nniform.  Hence 
there  are  two  kinds  of  eciuilibrium,  the  one  relating  to 
bodies  at  rest,  the  otiier  relating  to  bodies  in  motion.  The 
former  is  sometimes  called  Sialic  Equilibrium  and  the  lat- 
ter KtHelic  (or  Dynamic*)  Equilibrium.  The  problem  of 
sialics  is  to  delerminc  the  conditions  under  tvhich  forces  act 
tohen  they  keep  bodies  at  rest. 

27.  Concurring  and  Conspiring  Forces.— ResulV 

ant.— When  several  forces  have  a  common  point  of  appli- 
cation they  are  called  concni-l-ing  forces  ;  when  -they  act  at 
the  same  point  and  along  the  same  right  line  they  are 
called  conspiring  forces. 

The  resultant  of  two  or  more  forces  is  that  force  which 
singly  will  produce  the  same  effect  as  tlie  forces  them- 
selves when  acting  together.  The  individual  forces,  when 
considered  with   reference   to    this    resultant,   are    called 

•  Gregory's  Mochauics,  p.  U. 


22 


VOMPOSITIOX   OF  COMSl'lRIXO  FORCES. 


componenta.     1'Iie  process  of  finding  the  resultant  of  several 
forces  is  called  the  cumposi/ion  of  forces. 

28.  Composition    of  Conspiring  Forces.— Condi 
tion  of   Equilibrium. — When   two  or  more  conspiring' 
forces  act  in   the   same   direction,  it  is  evident  that  the 
resultant  force  is  eciual  to  their  sum,  and  acts  in  the  same 
direction. 

When  two  conspiring  forces  act  in  opposite  directions 
their  resultant  force  is  e<iual  to  their  dififereuce,  and  acts  iu 
the  direction  of  the  greater  component. 

When  several  conspiring  forces  act  in  different  directions 
tiie  resultant  of  the  forces  acting  in  one  direction  equals 
the  sum  of  these  forces,  and  acts  in  the  same  direction  ; 
and  so  of  the  forces  acting  in  the  oi)posite  direction. 
Therefore,  the  resultant  of  all  the  forces  is  equal  to  the 
difference  of  those  sums,  and  acts  in  the  direction  of  the 
greater  sum.  Hence,  if  the  forces  acting  in  one  direction 
are  reckoned  positive,  and  those  in  the  opppsite  direction 
negative,  their  resultant  is  equal  to  their  algebraic  sum  ; 
its  sign  determining  the  direction  in  which  it  acts.  Thus, 
if  F^,  Pg,  Pj,  etc.,  are  the  conspiring  forces,  some  of 
which  may  be  positive  and  the  others  negative,  and  li  is 
the  resultant,        have 


/<;  =  P,  +  Pg  +  P3  +  etc.  =  SP, 


(1) 


in  which  i:  denotes  the  algebraic  sum  of  the  terms  similar 
to  that  written  immediately  after  it. 

Cou.— The  condition  that  the  forces  may  be  in  equilib- 
rium is  that  their  resultant,  and  therefore  their  algebraic 
sum,  must  vanish.  Hence,  when  the  forces  are  in  equilil)- 
rium  we  must  have  7^  =  0  ;  therefore  (1)  becomes 


^1  +  ^8  +  i'a  +  t'tc.  =  1P  =  0. 


(^) 


nt  of  several 


s. — Condi 
conspiriiij,' 
nt  that  tlu' 
in  tbe  same 

e  directions 
and  acts  in 

t  directions 
ction  equals 
3  direction  ; 
)  direction, 
qual  to  the 
;tion  of  the 
le  direction 
te  direction 
braic  sum  ; 
cts.  Thus, 
s,  some  of 
3,  and  R  is 


(1) 


rms  similar 


in  equilib- 
ir  algebraic 
I  in  equilili- 
les 

(2) 


COMPOSITIOS  OF   VELOCITIES. 


23 


29.  CompoBitioii  of  Velocities.— // f*  particle  be 
moving  with  two  unifonii  velocities  represented  in 
magnitude  and  direction,  by  the  two  adjacent  sides 
of  a  parallelogram,  the  resultant  velocity  will  be 
repr  sented  in  magnitude  and  direction  by  the 
diagonal  of  t:.e  parallelogram. 

Let  the  particle  move  with  a  uniform 
velocity  v,  which  acting  alone  will  take 
it  in  one  second  from  A  to  B,  and  with 
a  uniform  velocity  v',  which  acting 
alone  will  take  it  in  one  second  from  A 
to  C  ;  at  the  end  of  one  second  the  par- 
ticle will  be  found  at  D,  and  AD  will  represent  in  magni- 
tude and  direction  the  resultant  of  the  velocities  represented 
by  AB  and  AC. 

Suppose  the  particle  to  move  uniformly  along  a  straight 
tube  which  starts  from  AB,  and  moves  uniformly  parallel 
to  itself  with  its  extremity  in  AC.  When  the  particle  starts 
from  A  the  tube  is  in  the  position  AB.  When  the  particle 
has  moved  over  any  part  of  AB,  the  end  of  the  tube  has 
moved  over  the  same  part  of  AC,  and  the  particle  is  on  the 

line  AD.    For  example,  let  AM  be  the  -th  part  of  AB,  and 

AN  be  the  -th  part  of  AC  ;  while  the  particle  moves  from 

n 
A  to  M,  the  end  A  with  the  tube  AB  will  move  from  A  to 
N,  and  the  particle  will  be  at  P,  the  tube  occupying  the 
position  NL,  and  PM  being  parallel  and  equal  to  AN.     P 
can  be  proved  to  be  on  the  diagonal  AD  as  follows  : 

AM  :  MP    ::    --  :  -^    "    AB  :  AC  (=..  BD); 
n        n 

therefore  P  lies  on  the  diagonal  AD.    Also  since 

AM  :  AB    : :    AP  :  AD, 


* 


24  COMPOSITION  OF  FORCES. 

the  resultant  velocity  is  uniform.  Hence,  the  diagonal  AD 
represents  in  niaj^niitude  and  direction  the  nsnltunt  of  the 
velocities  represented  by  AB  and  AC. 

Tins    i)roposition    is    known    as  the   Parallelogram   of 
Velocities. 

30.  Composition  of  rorces.— Prom  the  Parallelo- 
gram of  Velocities  the  Parallelogram  of  Forces  follows 
immediately.  Since  two  simultaneous  velocities,  AB  and 
AC,  of  a  particle,  result  in  a  single  velocity,  AD,  and  since 
these  three  velocities  may  be  regarded  as  the  measures  of 
three  separate  forces  all  acting  for  the  same  time  (Art.  HI), 
it  follows  that  the  effect  produced  on  a  particle  by  the  com- 
bined action,  for  the  same  time,  of  two  forces  nniy  be  pro- 
duced by  the  acti(}n,  for  the  same  time,  of  a  single  force, 
whicii  is  therefore  called  the  rexuttaut  of  the  other  two 
forces;  and  these  forces  are  repret^entod  in  magnitude  and 
direction  by  AB,  AC,  and  AD.  (See  Minchin,  p.  7,  also 
Garnett's  Dynamics,  p.  10.) 

Hence  if  two  concurring  forces  be  represented  hi  magni- 
iucle  and  direr/ion  bg  the  adjacent  sides  of  a  parallelogram, 
their  resultanl  will  be  represented  in  magnitude  and  direction 
by  the  diagonal  of  the  parallelogram.  Care  must  bo  taken 
in  constructing  the  parallelogram  of  forces  that  the  com- 
ponents both  act  from  the  angle  of  the  parallelogram  from 
which  the  diagonal  is  drawn. 

This  ])r<>|'ci.<ilioii  Iimh  Ix-eii  jjrovcd  in  varioun  wnys.  It  was  onun- 
cititoil  in  its  present  Cocin  by  Sir  Ihbiic  Newton,  and  by  Vnriprnon,  tlip 
ci'l(!l)rnti'd  nuitiicmuti'-ian,  in  the  year  1087,  i)robal)ly  indepiMidcnt  of 
eai'h  otlicr.  Since  thai  tinu^  various  proofs  ;>f  it  liavu  been  given  l)y 
ditf'creiit  niatlienuiticianH.  One  vorli  gives  a  (Jiiciisfioii,  more  or  less 
complete,  of  4')  otlier  proofs.     A   noted  analytic  proof  ]b  given  by 

M.  I\.i.s.son.     (See  Price's  Cal.,  Vol.  Ill,  p.  1!»).     H e  antliors  ol)jec! 

to  proving  the  parallelopani  of  forces  liv  nieanw  of  the  parallelogruni 
of  velocities.  (See  Oregory's  MechanicH.  p,  14.)  The  student  wli.i 
wants  other  proofs  is  referred  to  Duchayla's  proof  as  found  in  'I'od- 
hunter's  Statics,  p.  7,  uud  in  (ialbraith's  MuchunicB,  p.  7.  and  iu  many 


thia.xole  of  forces. 


25 


c  diagonal  AD 
sultunt  of  the 

aJleloyram  of 


he  Parallvlo- 
'•'orcps  follows 
ities,  AH  ami 
(VD,  uiul  since 
!  nieasnres  of 
me  (Art.  HI), 
e  by  the  coni- 
nniy  be  ])ro- 
singlc  force, 
he  other  two 
agnitudc  and 
in,  p,  7,  also 

'ed  hi  mofjni- 
mraUelogram, 
and  direction 
aust  bo  taken 
hat  the  coni- 
jlograni  from 


L  It  was  enun- 
7  Variprnon,  tlip 
iii(iii|)(>n^l('nt  of 

bt'fii  givnii  l)y 
1)11,  more  or  l<!s» 
of  is  givon  by 

autliofH  olycc! 
■  piirullclogriiiii 
<■  Htudcnt  who 
I  found  in  Tod- 
7,  and  iu  many 


other  works  ;  or  to  Liipliu-c's  proof.     (See  Mecanique  Celeste,  Liv.  I, 
chap.  1.) 

If  e  be  the  angle  between  the  sides  of  the  parallelogram, 
AB  and  AC  (Fig.  •^),  and  P  and  Q  represent  the  two  com- 
ponent forces  acting  at  A,  and  It  represent  the  resultant, 
AD,  we  have  from  trigonometry. 


Ri  =  P^+  Q'i  +  'iPQ  cose 


(1) 


an  equation  which  gives  the  mmjnUude  of  the  resultant  of 
two  forces  in  terms  of  the  magnitudes  of  the  two  forces  and 
the  angle  between  their  directions,  the  forces  being  repre- 
sented by  two  lines,  both  drawn  from  the  point  at  which 
they  act. 

Cou.— If  0  =  90°,  and  «  and  d  be  the  angles  which  the 
direction  of  A'  makes  with  the  directions  of  P  and  Q,  we 
have  from  (1) 

(2) 
(3) 


Also 


cos  a  = 


from  which  the  nuignitnde  and  direction  of  the  r-'sultant 
are  determined. 

31.  Triangle  of  Forces.—//  fhraft  eoncnrriti.^ 
forces  be  represented  hi  nut'Jiiitude  and  direct  ion. 
1>!I  the  sides  <)f  a  triangle,  taken  in,  order,  theij  will 
lie  in  eqnUiliriuni. 

Let  AHC  bo  the  triangle  whose 
sides,  taken  in  order,  represent  in 
nnignitude  and  direction  three  forces 
aiiplied  at   the  point  A.     Complete 


Fia.3 


^ 


26 


TRIANGLE  OF  FORCES. 


the  i)arallelograrn  ABCD.  Tlicn  tliL'  forces,  AB  and  BC, 
a])i)lied  at  A,  arc  cxjiressed  by  AB  and  AD  (since  AD  is 
equal  and  i)arallol  to  BC).  But  the  resultant  of  AB  and 
AD  is  AC,  acting  in  the  direction  AC.  Therefore  the  three 
forces  represented  by  AB,  BC,  and  CA,  are  equivalent  t.i 
two  forces,  AC  and  CA,  the  former  acting  from  A  toward 
C  and  the  latter  from  C  towards  A,  which,  being  equal  and 
o))i)ositc,  will  clearly  balance  each  otlur.  Therefore  the 
three  forces  represented  by  AB,  BC,  and  CA,  acting  at  the 
point  A,  will  be  in  equilibrium. 

It  should  be  observed  that  though  BC  represents  the 
maynitude  and  direction  of  the  component,  it  is  not  in  the 
line  of  its  action,  because  the  three  forces  act  at  the 
point  A. 

The  converse  of  this  is  also  true  ;  viz..  If  three  concurring 
forces  are  in  e(|uilibrium,  they  may  be  re])resented  in  mag- 
nitude and  direction  by  the  sides  of  a  triangle,  drawn 
l)ariill('l  respectively  to  the  directions  of  the  forces. 

Thus,  if  AB  and  BC  represent  two-forces  in  magnitude 
and  direction,  AC  will  represent  the  resultant,  and  hence  to 
produce  equilibrium  the  resultant  force  AC  must  be  opposed 
by  an  equal  and  opposite  force  CA.  Therefore,  the  three 
forces  in  equilibrium  will  be  represented  by  AB,  BC,  and 
CA. 

Cor.— When  three  concurring  forces  are  in  equilibrium, 
each  is  eipnil  and  directly  opposite  to  the  resultant  of  the 
other  two. 

32.  Relations  between  Three  Concurring  Forces 
in  Equilibrium.— Siuce  the  sides  of  a  plane  triangle  are 
as  the  sines  of  iiu>  opposite  angles,  we  have  (Fig.  I}) 

AB  :  BC  (or  AD)  :  A(;   ::   sin  WW  :  sin  WW  :  sin  ABC 

::   sin  T).\C  :  sin  WW  :  sin  MAI>. 

Hence,  calling  /',  Q,  and  A',  (he  forces  represented  by  AM. 
AD,  and  AC,  ami  denoting  the  angles  l)etween  the  diree- 


POLYGON    OF   FORCES. 


27 


ABand  BC, 
(since  AD  is 
it  of  AB  ami 
fore  the  three 
equivalent  to 
ni  A  toward 
ing  equal  and 
riierctbre  the 
acting  at  the 

opresents  the 
is  not  in  tlie 
i  act  at  the 

3e  concurring 
ited  ill  mag- 
mgle,  drawn 

•CCS. 

n  magnitude 
and  hence  to 
st  be  opposed 
re,  the  three 
AB,  BC,  and 

equilibrium, 
ultant  of  the 

ing  Forces 

)  triangle  are 

)  :  sill  AHC 
'  :  sin  MAI>. 

iited  l».v  AH. 
Ml  the  direc- 


(1) 


tions  of  the  forces  P  and  Q,  Q  and  h\  and  li  and  P,  by 

A      A  A 

I'Q,  QR,  and  RP,  respectively,  we  have 

A  A  a' 

sin  QR       sin  RP       sin  PQ 

Therefore,  when  three  coucurrim/  forces  are  in  cquilibriinn 
they  arc  respectively  in  the  same  proportion  as  t/te  sines  of 
the  a:'(jles  included  between  the  directions  of  the  other  two. 

33.  The  Polygon  of  Forces.—//"  an//  num/xir  of 
eoncivrring  forces  be  represented  in  magnitiulc  (ind, 
direction  hy  the  sides  of  a  closed  polygon  taken  in 
order,  they  will  he  in  equiUbriiun. 

Ijet  the  forces  be  represented  in 
magnitude  and  direction  by  the  lines 
AP,,  \\\,  AP3,  AP„  AP„.  Take 
AB  to  represent  AP,,  through  B  draw 
BC  equal  and  i)arallel  to  APg  ;  the 
resultant  of  the  forces  AB  and  BC,  or 
AP,  and  APg  is  represented  by  AC 
(Art.  ;n).  Of  course  the  force,  BC, 
acts  at  A  and  is  parallel  to  liC.  Again  through  C  draw  CD 
equal  and  parallel  to  APj,,  the  resultant  of  AC  and  CD,  or 
AP,,  APj,  and  AP3  is  AD.  Also  through  D  draw  DE 
equal  and  parallel  to  AP^,  the  resultant  of  AD  and  DE,  or 
AP,,  APg,  AP3,  and  AP4  is  AE.  Now  if  AE  is  equal  and 
oi)i)(»8ite  to  APg  the  system  is  in  equilibrium  .(Art.  IH). 
Hence  the  forces  represented  by  AB,  BC,  CI),  DE,  EA 
will  l)e  in  eciuilibrium. 

(joK.  !.— Any  one  side  of  the  pi»lygon  represents  in 
magnitude  and  direelion  tlie  resiiKunt  of  ail  tiie  forces 
represented  l»y  the  remaining  sides. 

(!,)i{.  •).-_ir  the  lines  representing  the  forces  do  iiol  I'onii 
n  closed  polygon  the  forces  are  not  in  oquilibriuiu  ;  \\\  this 


Fia.4 


28 


PAKALhKLOPIl'ED    OF   FORCES. 


case  the  last  side,  AE,  taken  from  A  to  Yj,  or  tliut  which  i? 
required  to  close  up  the  polygon,  represents  in  magnitude 
aiui  direction  tlie  resultant  of  the  system. 

34.  Farallelopiped  of  Forces. — //  three  concur- 
ring forces-,  not  in  the  same  plane,  are  represcntvil 
in  nia.gnitade  and  direction  by  the  three  edges  of 
a  parallelopipcd,  then  tlie  resultant  will  be  repre- 
sented in  magnitude  and  direction  by  the  diag- 
onal;  conversely,  if  the  diagonal  of  a  parallel - 
opiped,  represents  a  force,  it  is  eqaivalent  to  three 
forces  represented  by  the  edges  of  the  p'lraUel- 
opiped. 

Let  the  three  edges  AB,  AC,  AD  of  tho 
parallelopii)ed  represent  the  three  forces, 
applied  at  A.  Then  the  resultant  of  the 
forces  AH  and  AC  is  AE,  the  diagonal  of 
the  face  AHCE;  and  tho  resultant  of  the 
forces  AE  and  AD  is  AF,  the  diagonal  of 
the  parallelogram  ADFE.  Hence  AF  represents  tlie 
resultant  of  the  three  forces  AB,  AC,  and  AD. 

Conversely,  tlic  force,  AF,  is  equivalent  to  the  three 
components  AB,  AC,  aiul  AD. 

Let  1\  Q,  S  represent  the  three  forces  AB,  AC,  AD;  K, 
the  resultant;  «,  (i,  y,  the  angles  which  the  direction  of  E 
nuikes  with  tlie  directions  of  /',  Q,  S,  and  sujjpose  the 
forces  to  act  at  right  angles  with  each  other.     Then  since 

AF"  =  A\f  +  AC*  +  Ai)^ 
P 

ir 

Q 

li'l 


we  have 
also, 


cos  «  = 


cos  [i  -: 


(1) 


(2) 


cosy  =  y-, 


tliut  which  i? 
in  magnitudi' 


rrec  conctvr- 
represeiifi'd 
*'ee  edges  of 
ill  be  repre- 
uj  the  dUig- 
a  fjd.ralliil- 
eiU  to  three 
le  jM'jrallel- 


^    Fig.5 

(presents    the 

to   the   tlirec 

AC,  AD ;  /.', 
ircctioii  of  li 
sniiposo   the 
Thou  since 

(1) 


(2) 


BXAMfLES. 


29 


from  which  the  magnitude  and  tlireotion  of  the  resultant 
are  determined. 

EXAMPLES. 

1.  Three  forces  of  5  lbs,,  ;}  lbs.,  and  2  lbs.,  respectively, 
act  upon  a  point  in  thesaine  direction,  and  two  otber  forces 
of  8  lbs.  and  1)  lbs.  act  in  the  0})posite  direction.  Wbat 
single  force  will  keep  the  point  at  rest?  Anti.  7  lbs. 

2.  Two  forces  of  5^  lbs.  and  31  ll)s.,  applied  at  a  point, 
urge  it  in  one  direction ;  and  a  force  of  2  Il)s.,  applied  at 
the  same  point,  urges  it  in  the  opi)ositc  direction.  What 
additional  force  is  necessary  to  preserve  equilibrium  ? 

Ans.  7  lbs. 

3.  If  a  force  of  13  lb?,  be  represented  by  a  line  of  6| 
inches,  what  line  will  represent  a  force  of  T|  lbs.? 

Ans.  3 J  inches. 

4.  Two  forces  whose  magnitudes  are  as  3  to  4,  acting  on 
a  point  at  right  angles  to  each  other,  produce  a  resultant  of 
20  lbs.;  required  the  component  forces. 

A71S.  12  lbs.  and  16  lbs. 

5.  Let  ABC  be  a  triangle,  and  D  the  middle  point  of 
the  side  BC.  If  the  three  forces  represented  in  magnitude 
and  direction  by  AB,  AC,  and  AD,  act  ui)on  the  point  A; 
find  the  direction  and  magnitude  of  the  resultant. 

Ans.  The  direction  is  in  the  lino  AD,  and  the  magni- 
tude is  represented  by  3AD. 

^,  When  r  =  Q  and  0  =  00°,  find  /?. 

7.  When  J'  =  Q  and  0  =  I35^  find  Ji.  

Ans.   A'  =  pVl-  V2. 


8.    When  P  =  Q  and  0 


12(r,  find  A'. 

vl//.v.   /.'  ^  P. 


^^ 


30 


RESOLVtlON    OF   FOltOES. 


J 


9.  If  P  =  q,  sliow  tl.at  thoir  resultant  R  =  ^P  cos  f- 
hS).  If  /'  =  8,  and  Q  =  10,  and  6  =  GO",  find  Jx\ 

A)ifi.  Ji  =  i  Vai. 

11.  If  P  =  144,  R=  145,  and  6  =  90°,  find  Q. 

A71S.   Q  =  17. 

12.  Two  forces  of  4  lbs.  and  3  \/2  lbs.  act  at  an  angle  of 
45°,  and  a  third  force  of  Vi^  lbs.  acts  at  right  angles  to 
their  plane  at  the  same  point ;  find  thoir  resultant. 

Ans.  10  lbs. 

35.  Resolution  of  Forces.— By  the  resolution  of  forces 
is  vieant  the  process  of  finding  the  components  of  given  forces. 
We  have  seen  (Art.  30)  that  two  concurring  forces,  P  and 
^  =:  AB  and  AC,  (P'ig.  2)  are  equivalent  to  a  single  force 
R  =  AD ;  it  is  evident  then  that  the  single  force,  R,  acting 
along  AD,  cau  bo  replaced  by  the  two  forces,  P  and  Q, 
represented  in  magnitude  and  direction  by  two  adjacent 
sides  of  a  parallelogram,  of  whicii  AD  is  the  diagonal. 

Since  an  infinite  number  of  parallelograms,  of  each  of 
which  AD  is  the  diagonal,  can  be  constructed,  it  follows 
that  a  single  force,  R,  can  be  resolved  into  two  other  forces 
in  an  infinite  number  of  ways. 

Also,  each  of  the  forces  AB,  AC,  may  be  resolved  into 
two  others,  in  a  way  similar  to  that  by  which  AD  was 
resolved  into  two ;  and  so  on  to  any  extent.  Hence,  a  single 
force  may  be  resolved  into  any  number  of  forces,  whose 
combined  action  is  equivalent  to  the  original  force. 


Cor. — The  most  convenient  compo- 
nents into  which  a  force  can  be  resolved 
are  those  whose  directions  are  at  right 
angles  to  each  other.  Tiius,  let  OX 
and  OF  be  any    two    lines    at  right 


Fig.6 


'1» 


angles  to  each  other,  and  P  any  force  acting  at  0  in  the 


MAGNITUDE    AND    DIRECTION    OF   HESVLTaNT.       31 


:  2P  COS  I- 

find  li. 
=  2  Vol. 

nd  Q. 
Q  =  17. 

an  angle  of 

lit  angles  to 

ant. 

IS.  10  lbs. 

Hon  of  forces 
given  forces. 
)rces,  P  and 
single  force 
2c,  R,  acting 
IS,  P  and  Q, 
wo  adjacent 
agonal. 
,  of  each  of 
1,  it  follows 
other  forces 

'esolved  into 
ich  AD  was 
ince,  a  single 
jrces,  whose 
roe. 


Fig.6  M 

at  0  in  the 


plane  XOY.  Then  complpting  the  rectangle  OMPN  wc 
find  the  components  of  /'  along  the  axes  OA' and  0  V  to  be 
OM  Awi  ON,  which  denote  by  X  and  Y.  Then  we  have 
clearly 

(1) 


X=  P  cos 


Y  =  P  sin 


«; ) 


vhcre  «  is  the  angle  which  the  direction  of  P  makes  with 
OX.  These  components  X  and  Y  are  called  the  rect- 
angular components.  The  rectangular  component  of  a 
force,  P,  along  a  right  line  is  Px  cosine  of  angle  between 
line  and  direction  of  /'. 

In  strictness,  when  we  speak  of  the  component  of  a  given 
force  along  a  certain  line,  it  is  necessary  to  mention  the 
other  line  along  which  the  other  component  acts.  In  this 
work,  unless  otherwise  expressed,  the  component  of  a  force 
along  any  line  will  be  understood  to  be  its  rectangular 
component;  i.e.,  the  resolution  will  be  made  along  this  line 
and  the  line  perpendicular  to  it. 

36.  To  find  the  Magnitude  and  Direction  of  the 
Resultant  of  any  number  of  Concurring  Forces  in 
one  Plane. — When  there  are  several  concurring  forces,  the 
condition  of  their  equilibrium  may  be  expressed  as  in 
Art.  33,  Cors.  1  and  2.  But  in  practice  we  obtain  much 
simpler  results  by  using  the  principle  of  the  Resolution  of 
Forces  (Art.  35),  than  those  given  by  the  principle  of 
Composition  of  Forces. 

Let  0  be  the  point  at  which  all 
t  lie  forces  act.  Through  0  draw  the 
rectangular  axes  XX',  YY'.  Let 
/',,  Pg,  P3,  etc.,  be  the  forces  and 
«!,  «2,  «3,  etc.,  be  the  angles  which 
their  directions  make  with  the  axis 
of  X. 

Now  resolve  each  force  into  its  two 
components  along  the  axes  of  x  and  y. 


p 

Y 

^B 

R 

xl- 

^ 

X 

-j:. 

f 

0            X, 

y> 

Y' 

f 

7 
• 

Fig.7 

Then   the  com- 


^M 


32       MAGMTVbE    AXD    DIRECTtoy    Of  RESULTANT. 


ponciits  along  the  axis  of  .r  (^--components)  are  (Art. 
35,  Cor.),  Tj  cos  «,.  I\  cos  «2,  P^  cos  «3,  etc.,  and  those 
along  the  axis  of  y  are  I\  sin  «,,  /'^  sin  (t^,  i'j  sin  a^, 
etc.;  and  therefore  if  Xand  F denote  the  algebraic  sum  of 
the  x-components  and  y-conipoaents  respectively,  we  have 


X  =  Pj  cos  «i  + 1\  cos  «j  +  P3  cos  «3  -f  etc. 
=  "LP  cos  a, 


(2) 


P"=  P,  sin  rtj+Pg  sinag  +  Pj  sin  Wj+etc.  |^ 
=  }LP  sin  «.  ( 

Let  R  be  the  resultant  of  all  the  forces  acting  at  0,  and  0 
the  angle  which  it  makes  with  the  axis  of  x ;  then  resolving 
R  into  its  x-  and  y-compononts,  we  have 


P  cos  0  =  X  =  SP  cos  «, ) 
Rs\nd=  Y  —  SP  sin  «.  j 

.    7?  =  Jr2+F2;  tanO 


F 
X' 


(3) 


(4) 


which    determines  the  magnitude  and  direction  of    the 
resultant. 

Sen. — Regarding  OX  and  OY  as  positive  and  OX/^  and 
OF*  as  negative  as  in  Anal.  Geora.,  we  see  that  Ox^,  Oy^, 
Oy^  are  positive,  and  Ox^,  Ox^,  Oy^  are  negative.  Tlic 
forces  may  always  be  considered  as  positive,  and  hence  the 
signs  of  the  components  in  (1)  and  {'i)  will  be  the  same  as 
those  of  the  trigonometric  functions.  Thus,  since  «g  is 
>  90°  and  <  180°  its  sine  is  positive  and  cosine  is  negative; 
since  «3  is  >  180°  and  <  270°  both  its  sine  and  cosine  are 
negative. 

37.  The  Conditions  of  Equilibrium  for  any  number 

of  Concurring  Forces  in  one  Plane.— For  the  eqnilil)rinni 

of  tile  forces  we  must  have  R  —  0.     Hence  (4)  of  Art.  ',)*> 

becomes 

X2+  }'»  =  0.  (1) 


}Ljm 


TLTANT. 

)  are  (Art. 
c,  and  those 
2,  i'j  sin  «3, 
>braic  sum  of 
ily,  we  have 

+*•[   (1, 

;  at  0,  and  6 
lien  resolving 

(3) 

(4) 
ition  of    the 

md  0Z»  and 
lat  Oa;,,  Oy,. 
gative.  Tlic 
id  hence  tlie 
the  same  as 
1,  since  «g  is 
e  is  negative; 
id  cosine  are 

my  number 

eeqnilibriuni 
I)  of  Art.  ;j(; 

(1) 


EXAMPLES. 


88 


Now  (1)  cannot  be  satisfied  so  long  as  X  and  Y  are  real 
quantities  unless  X=0,  V  =0;  therefore, 

X  =  ^P  cos  «  =  0  and  r  =  IP  sin  «  =  0.        (2) 

Hence  these  are  the  two  necessary  and  sutlicient  conditions 
for  the  equilibrium  of  tlie  forces;  that  is,  //w  algebraic  sum 
of  the  rectangular  components  of  the  forces,  along  each  of 
two  right  lines  at  right  angles  to  each  other,  in  the  plane  of 
the  forces,  is  equal  to  zero.  As  the  conditions  of  equilibrium 
must  be  independent  of  the  system  of  co-ordinate  axes,  it 
follows  that,  if  any  number  of  concurring  forces  in  one 
plane  arc  in  equilibrium,  the  algebraic  sum  of  the  rectan- 
gular components  of  the  forces  along  every  right  line  in  their 
plane  is  zero. 

EXAMPLES. 

1.  Given  four  equal  concurring  forces  whose  directions 
are  inclined  to  the  axis  of  x  at  angles  of  15°,  75",  135°, 
and  325°  ;  determine  the  magnitude  and  direction  of  their 
resultant. 

Let  each  force  be  equal  to  P  ;  then 

X  -  P  cos  15°  +  P  cos  75°  +  P  cos  135°  +  P  cos  225° 

.3^-2 


=  P 


2* 


r  =  P  sin  15°  +  P  sin  75°  +  P  sin  135°  +  P  sin  225° 
=  P(#. 

.-.    i2  =  P(5-2v'3)*- 

tan  9  =  -T • 

3* -2 

2.  Given  two  equal  concurring  forces,  P,  whose  direc- 
tions are  inclined  to  the  axis  of  x  at  angles  of  30°  and  315°; 
find  their  resultant.  Ans.  i?  =  1.59  P. 

a* 


^^ 


34 


COiWVltU'IXa    FOIICES. 


3.  Given  throe  concnrriiig  forces  of  4,  5,  and  6  lbs., 
wlioso  directions  aro  inclinc'<]  to  the  axis  of  x  ni  angles  o.' 
0°,  60°,  and  135'  rcspectiwlv  ;  find  tiieir  resultant. 

Afis.  n  =  Vu7  +  15  Vc  —  39  Va. 

4.  Given  three  ecpial  coiieiirrin<?  forws,  P,  wliose  direc- 
tions are  inclined  to  the  axis  of  x  at  angles  of  30°,  G0°,  and 
105°  ;  find  their  resultant.  J«.s.  72  =  1.07  P. 

^  5.  Given  three  concurring  forces,  100,  50,  and  '^00  lbs., 
whose  directions  are  inclined  to  the  axis  of  x  at  angles  of 
0°,  00°,  and  180° ;    find  the  magnitude  and   direction   of 


their  resultant. 


Ans.  R  =  86.6  lbs. :  6  =  150° 


38.  To  find  the  Magnitude  and  Direction  of  the 
Resultant  of  any  number  of  Concxurring  Forces  in 

Space.— Let  P^,  P^,  P^,  etc.,  be  the  forces,  and  the 
whole  be  referred  to  a  systoni  of  rectangular  co-ordinates. 
Ijet  «i,  /3,,  yj,  be  the  angles  whicli  the  direction  of  P^ 
makes  with  three  rectangular  axes  drawn  through  the  point 
of  application  ;  let  w^,  (3^,  y^,  be  the  angles  v  Inch  the  direc- 
tion of  Pg  makes  with  the  same  axes  ;  «3,  (3^,  y^  ';he 
angles  which  Pj  makes  with  the  same  axes,  etc.  Resolve 
these  forces  along  tlie  co-ordinate  axes  (Art.  35) ;  the  com- 
ponents of  P,  along  the  axes  are  P,  cos  «,,  Pj  cos  3,,  P, 
cos  yj.  Resolve  each  of  the  other  forces  in  the  same  way, 
and  let  X,  V,  Z,  be  the  algebraic  sums  of  the  components 
of  the  forces  along  the  axes  of  x,  y,  and  z,  respectively ; 
then  we  have 

X  =  Pj  cos  «i  -H  Pg  cos  a,  -}-  Pj  cos  «3  -f-  etc. 

=  £P  cos  «. 
Y—P^  cos  01  +  Pj  cos  /Sg  +  Pg  co8 /Jj  +  etc.l  .  . 

—  SP  cos  /3.  /  ^^^ 

iT  =  P,  COS  yi  4-  Pj  cos  y^  -f  P,  cos  yj  +  etc. 

=  iP  cos  y. 


CONDinoNS   OF  EQVITAUKIVM. 


35 


and  6  lbs., 
id  tingles  d," 
ant. 

-  39  Va. 

tfliose  (liroc- 
[)°,  G0°,  and 
=  1.07  P. 

id  '^OC  lbs., 
it  angles  of 
lircction   of 
=  150°. 


ion  of  the 
Forces  in 

's,  and  tlie 
3-ordinates. 
tion  of  Pj 
;li  the  point 
h  the  direc- 

c.  Resolve 
;  the  com- 
cos  3,,  P, 
same  way, 
omjionents 
spectively  ; 


+  etc. 
+  etc.( 
+  etc. 


(1) 


Let  R  be  the  resultant  of  all  the  forces;  and  let  the 
aiiirles  which  its  direction  makes  with  tlie  three  axes  be  a. 
b.  r  ;  then  as  the  resolved  parts  of  I!  along  flie  tliree  c'l-or- 
dinate  axes  are  equal  to  the  sum  of  tlie  resolved  parts  of 
the  several  components  along  the  same  axes,  we  liave 

R  coaa  =  X,     R  cos  b  =  V,     R  cos  c  =  Z.       (2) 


Stpiaring,  and  adding,  we  get 


COS  a  =  T.  >     cos  0  =  -TT ,     cos  p  = 
A  h 


Z 

R' 


(3) 

(4) 


which  detei-mines  the  magnitude  of  the  resultant  of  any 
system  of  forces  in  space  and  the  angles  its  direction  makes 
with  three  rectangular  axes. 

39.  The  Conditions  of  Equilibrium  for  any  num- 
ber of  Concurring  Forces  in  Space.— If  the  forces  are 

in  equilibrium,  R  =  0  ;  therefore  (3)  of  Art.  38  becomes 

X«  +  F«+  Z^  =  0. 

But  as  every  s(iuare  is  essentially  positive,  this  cannot  be 
unless  .r  =  0,  Y  =  0,  Z  =  0  ;  and  therefore 

SP  cos  «  =  0,     SP  cos  i3  =  0,     SP  cos  y  =  0 ;     (1) 

and  these  are  the  conditions  among  the  forces  that  they 
may  be  in  equilibrium  ;  that  is,  tlie  sum  of  the  components 
of  the  forces  along  each  of  the  three  co-ordinate  axes  is 
equal  to  zero. 

40.  Tension  of  a  String. — By  the  tension  of  a  string 
is  meant  the  pull  along  its  fibres  which,  at  any  point,  tends 
to  stretch  or  break  the  stiing.  In  the  application  of  the 
preceding  principles  the  string  or  cord  is  often  used  as  a 


^ 


36 


EXAMPLES. 


means  of  communicating  force.  A  string  is  said  to  be  per- 
fectly flexible  when  any  force,  however  small,  wliich  is 
applied  otherwise  than  along  the  directitm  of  the  string, 
will  change  its  form.  In  this  work  the  string  will  he 
regarded  as  perfectly  flexible,  inextensil)le,  and  without 
weight. 

If  such  a  string  be  kept  in  equilibrium  by  two  forces, 
one  at  each  end,  it  is  clear  that  these  forces  must  be  equal 
and  act  in  opposite  directions,  so  that  the  string  assumes 
the  form  of  a  straight  line  in  the  direction  of  the  forces. 
In  this  case  the  tension  of  the  string  is  the  same  through- 
out, and  is  measured  by  the  force  applied  at  one  end  ;  and 
if  It  passes  over  a  smooth  peg,  or  over  any  number  of 
smooth  surfaces,  its  tension  is  the  same  at  all  of  its  points. 
If  the  string  should  be  knotted  at  any  of  its  points  to  other 
strings.  Me  must  regard  its  continuity  as  broken,  and  the 
tension,  in  this  case,  will  not  be  the  same  in  the  two  por- 
tions which  start  from  the  knot. 


EXAMPLES, 

1.  A  and  B  (Fig.  8)  are  two  fixed  a 
points  in  a  horizontal  line ;  at  A  is 
fastened  a  stiing  of  length  b,  with  a 
smooth  ring  at  its  other  extremity,  C, 
through  which  passes  another  string  with 
one  end  fastened  at  B,  the  other  end  of 


Fi9-8 


which  is  att;  ched  to  a  given  weight  W  ;    it  is  required  to 
determine  tiie  position  of  C. 

Before  setting  about  the  solution  of  statical  problems  of 
this  kind,  the  student  will  clear  the  ground  before  him,  and 
greatly  simplify  his  labor  by  asking  himself  the  following 
questions  :  (1)  What  lines  are  there  in  the  figure  whose 
lengths  are  already  given  ?  (2)  What  forces  are  there 
whose  magnitudes  are  already  given,  and  what  are  tlie 
Torces   whose   magnitudes  are  yet   unknown?    (3)  What 


id  to  be  per- 
il, which  is 
r  the  string, 
ring  will  he 
md    without 

J  two  forces, 
ist  be  equal 
iug  assumes 
'  the  forces, 
me  through- 
10  end  ;  and 
•  number  of 
f  its  points, 
ints  to  other 
ieu,  and  the 
the  two  por- 


Fig.8 

required  to 

problems  of 
ore  him,  and 
le  following 
figure  whose 
!8  are  there 
hat  are  the 
'    (3)  What 


EXAMPLES. 


37 


variable  lines  or  angles  in  the  figure  would,  if  they  were 
known,  determine  the  required  position  of  C  ? 

Now  in  this  problem,  (1)  the  linear  magnitudes  whicii 
are  given  are  the  lines  AB  and  AC.  (2)  The  forces  acting 
at  the  point  C  to  keep  it  at  rest  are  the  weight  W,  a  ten- 
sion in  the  string  CIJ,  and  anotiier  tension  in  tiio  string 

CA.  Of   tliese    W   is  given,    and   so   is   the   tension    in 

CB,  which  must  also  be  equal  to  W,  since  the  ring  is 
smooth  and  the  tension  therefore  of  WCB  is  the  same 
throughout  and  of  course  equal  to  W.  But  iis  yet  there  is 
nothing  determined  about  the  magnitude  of  the  tension  in 
OA.  And  (;3)  the  angle  of  inclination  of  the  string  CA  to 
the  horizon  would,  if  known,  at  once  determine  the  posi- 
tion of  C.  For  if  this  aii  "le  is  known,  we  can  draw  AC  of 
the  given  length;  then  j miing  C  to  B,  the  position  of  the 
system  is  completely  knowi>. 

Let  AB  =  rt,  AC  =  b,  CAB  =  d,  CBA  =  ^,  and  the 
tension  of  the  string  AC  =  T.  Then,  for  the  equilibrium 
of  the  point  C  under  the  action  of  the  three  forces,  W,  W, 
and  T,  we  apply  (2)  of  Art.  37,  and  resolve  the  forces 
horizontally  and  vertically :  and  equate  those  acting  towards 
the  right-hand  to  those  acting  towards  the  left ;  and  those 
acting  upwards  to  those  acting  downwards.  Then  the 
horizontal  and  vertical  forces  are  respectively 

IFcos<A  =  rcosS; 

PTsin^  -H  T'sinO  =  W. 

Eliminating  jTwe  have 

cos  6»  =  sin  (0  -f  ^)  ; 

.  • .    20  +  0  =  90°. 

Also,  from  trigonometry  we  have 

sin  {0  +  0)  _  a  ^ 
sin  0       ~~  6  ' 


(1) 
(a) 


ite 


38 


EXAMPLES. 


from  (1)  and  (2)  0  and  <p  may  be  found  ;  and  therefore  T 
may  be  found;  and  thus  all  the  circumstances  of  the 
problem  are  determined. 

2.  One  end  of  a  string  is  attached  to 
a  fixed  point,  A,  (Fig.  9) ;  the  .string,  after 
passing  over  a  smooth  i)eg,  B,  sustains  a 
given  weight,  P,  at  its  other  extremity, 
and  to  a  given  point,  C,  in  the  string  is 
•;notted  a  given  weight,  W.  Find  the  posi- 
tion of  equilibrium. 

Thi'  entire  length  of  the  string,  ACBP,  is  of  no  conse- 
quence, since  it  is  clear  that,  once  equilibrium  is  estab- 
lished, P  might  be  suspended  from  a  point  at  any  distance 
whatcvev  from  B.  The  forces  acting  at  the  jroint,  C,  are 
the  given  weight,  W,  the  tension  in  the  string,  CB,  which, 
since  the  peg  is  smooth,  is  P,  and  the  tension  in  the  string 
CA,  which  is  unknown. 

Let  AB  =z  a,  AC  =  b,  CAB  =  e,  CBA  ^  0,  and  the 
tension  of  the  string,  AC  =  T.  Then  for  the  ecjuilibrium 
of  the  point  C,  we  have  (A;-t.  32), 


cos  0 


W  ~  sin  (0  +  0) ' 

also,  from  the  geometry  of  the  figure,  we  have 

b  sin  {0  +  (j>)  =  a  sin  ((>. 
From  (1)  and  (2)  we  get 

P  _  3^  cos  e 

W  ~  n  sin  <^' 


(1) 


(2) 


or 


sin  0  =       --  cos  6; 


cos  <p 


_  A/ffia/^S!  -  AW*  C08»  6> 


aP 


therefore  T 
nces  of  the 


(f  no  consc- 
iin  is  estab- 
iiiy  distance 
|)oiiit,  0,  are 
CB,  which, 
II  the  string 

(/),  and  the 
equilibrium 


(1) 


(2) 


EXAMPLES.  89 

Expanding  sin  {0  +  0)  in  {'i),  and  substituting  in  it  these 
values  of  sin  (j>  and  cos  (p,  and  reducing,  we  have  the 
equation 

eos3  e ^., -^  cos^  0  +  -^^^  =  0, 

from  which  d  may  be  found.  (See  Minchin's  Statics, 
p.  29.) 

3.  If,  in  the  last  example,  the  weight,  W,  instead  of 
being  knotted  tn  the  string  at  C,  is  suspended  from  a 
smooth  rin/j  whioJi  is  at  liberty  to  slide  along  the  string, 
AOB,  find  the  position  of  equilibrium. 

W 

Atis.  sm  6  =  —p. 

41.  Equilibrium  of  Concurring  Forces  on  a 
Smooth  Plane. — If  a  particle  be  kept  at  rest  on  a  smooth 
surface,  plane  or  curved,  by  the  action  of  any  number  of 
forces  applied  to  it,  the  resultant  of  these  forces  must  be  in 
the  directioh  of  the  normal  to  the  surface  at  the  point 
where  the  particle  is  situated,  and  must  be  equivalent  to 
the  pressure  wiiicli  the  surface  sustains.  For,  if  the 
resultant  had  any  other  direction  it  could  be  resolved  into 
two  components,  one  in  the  direction  of  the  normal  and  the 
otiier  in  the  direction  of  a  tangent ;  the  first  of  these  would 
be  ojjposed  by  the  reaction  of  the  surface ;  the  second  being 
unopposed,  would  cause  the  ])article  to  move.  Hence,  we 
may  dispense  with  the  plane  altogether,  and  regard  its 
normal  reaction  as  one  of  the  forces  by  which  the  particle 
is  kept  at  rest.  Therefore  if  the  particle  on  which  the 
statical  forces  act  be  on  a  smooth  plane  surface,  the  case  is 
the  sanH>  as  that  treated  in  Art.  ;{!!,  viz.,  equilibrium  of  a 
particle  acted  upon  l>y  any  number  of  forces;  ami  in  writ- 
ing down  tiie  e(|uati(jiis  of  ecpiilibriiim,  we  merely  have  to 
include  the  normal  reaction  of  the  plane  among  all  the 
others. 


^ 


40 


KXAMPIjES. 


Fig. 10 


EXAMPLES. 

1.  A  heavy  particle  is  placed  on  a 
Bmooth  inclined  plane,  AB,  (Fig.  10), 
and  is  sustained  by  a  force,  /',  which 
acts  along  AB  in  the  vertical  plane 
Avhich  is  at  right  angles  to  AB ;  find 
P,  and  also  the  pressure  on  the  in- 
clined plane. 

The  only  effect  of  the  inclined  plane  is  to  produce  a 
normal  reaction,  R,  on  the  particle.  Hence  if  wc  intro- 
duce this  force,  we  may  imagine  the  plane  removed. 

Let  W  be  the  weight  of  the  ])artic)e,  and  a  the  inclina- 
tion of  the  plane  to  the  horizon. 

Resolving  the  forces  along,  and  perpendicular  to  Ali, 
since  the  lines  along  which  forces  may  be  resolved  ai'e 
arbitrary  (Art.  37),  we  have  successively. 


V  sin  «  =  0,     or    P  =  fF  sin  a  j 


and 


R  —  ]y  COS  «  =  0,     or     It  =  W  cos  «. 


If,  for  example,  the  weight  of  the  particle  is  4  oz.,  and 
the  inclination  of  tiie  plane  30",  there  will  be  a  normul 
pressure  of  2\/3  oz.  on  the  plane,  and  the  force,  /',  will 
be  'i  oz. 

a.  In  the  previous  example,  if  P  act  horizontally,  find 
its  magnitude,  and  also  that  of  R. 

Uesolving  along  AB  and  per|)endicular  to  it,  we  have 
successively, 

P  cos  «  —  H^sin  «  =  0.     or    /'  =  W  tan  «  ; 

W 


and     P  sin  a  +  W  cos  «  —  R  —  0,    .•.    R  — 


COB  a 


0  produce  a 
if  wc  intro- 
jved. 
the  inclina- 

ular  to  Ali, 
resolved  mv 


a 


is  4  07.., 

and 

ye  a  normal 

orce,  r, 

will 

on  tally, 

find 

it,  wo 

have 

tan  rt ; 

W 

UOB  a 


COADITIOAS  OF  JiQUILIIHilUM. 


41 


3.  If  the  particle  is  sustained  by  a  force,  P,  making  a 
given  angle,  0,  with  the  inclined  plane,  find  the  magnitude 
of  this  force,  and  of  the  i)ressure  on  the  plane,  all  the  forces 
acting  in  the  same  vertical  plane. 

Resolving  along  and  per|)endicular  to  the  plane  succes- 
sively, we  have 

P  cos  0  —  W  sin  «  =  0, 

and  R  -\-  P  sind  —  Wcostt  =  0, 

from  which  we  obtain 

P  =  W~^-     R  =  n/^'oM«  +  ^), 
cos  0'  cos  0      ' 

Rem. — The  advantage  of  a  judicious  selection  of  direc- 
tions for  the  resolution  of  the  forces  h  evident.  By  resolv- 
ing at  riglit  angles  to  one  of  the  unknown  forces,  wo 
obtiiin  an  etiuation  free  from  that  force ;  whereas  if  the 
directions  arc  .seleelod  at  random,  all  of  the  forces  will 
enter  each  ecjualion,  which  will  nuike  the  solution  less 
simple. 

The  student  will  observe  that  tiiese  values  of  /'  and  R 
could  have  been  obtained  at  once,  without  resolution,  by 
Art.  32. 

42.  Conditions  of  Equilibrium  for  any  number  of 
Concurring  Forces  when  the  particle  on  which  they 
act  is  Constrained  to  Remain  on  a  Given  Smooth 
Surface.  —  If  a  particle  be  ke])t  at  rest  on  a  smooth  sur- 
face by  tlie  action  of  any  number  of  forces  a|)plied  to  it, 
the  resultant  of  these  forces  must  be  in  the  direction  of  tho 
normal  to  the  surface  at  the  point  where  the  particle  ia 
siluiitcd.  iinil  must  i)e  equivalent  to  the  pressure  which  the 
surface  sustains  (Art.  41).  Hence  siiu-e  the  resultant  is  in 
the  direction  of  the  normal,  and  is  destroyed  by  the  roac- 


mm 


42 


CONDITIONS  OF  EQUILIBRIUM. 


tion  of  the  surface,  we  miiy  regard ,  this  reaction   as  an 
additional  force  directly  opposed  to  the  normal  force. 

Let  ^Vbe  the  normal  reaction  of  the  surface,  and  a,  0,  y, 
the  angles  which  JV  makes  with  the  co-ordinate  axes  of  ;/■, 
y,  and  z,  respectively.  Let  X,  V,  Z,  be  the  sum  of  the 
components  of  all  the  other  forces  resolved  parallel  to  tiu' 
three  axes  respectively.  The  reaction  iV^may  be  considered 
a  new  force,  which,  with  the  other  forces,  keeps  the  parti- 
cle in  equilibrium.  Therefore,  resolving  N  parallel  to  the 
three  axes,  we  have  (Art.  39), 


r-f-  iVcoLj3  = 

Z  ^  iVcos 


«  =  0,  ) 

r  =  0. ) 


(1) 


Let  u  =zf{x,  y,  z)  =  0,  be  the  cqua^^ion  of  the  given 
surface,  and  x,  y,  z  the  co-ordinates  of  the  f  iirticle  to 
whicii  the  forces  are  applied.  We  have  (Aial.  Geom., 
Art.  175), 

a' 


cos  a  = 

cos  i3  = 
cos  y  = 


V«'2  +  b'^  +  1 ' 
V 

_1^ 


(2) 


whore  a'  and  V  are  the  tangents  of  the  angler  which  the 
projections  of  the  normal,  N,  on  the  co-ordinate  i)lanes  xz 
and  yz  make  with  tlie  axis  of  z.  Si'.ice  the  nornud  is  per- 
pendicular to  the  plane  tangent  t«',  the  surface  at  {x,  y,  z). 
the  projections  of  the  normal  are  perpendicular  to  the 
traces  of  the  jdano.  Therefore  'Anal.  Geom.,  Art.  27. 
Cor.  1),  wo  have 

1  +  aa'  =  0,  (;}) 


and 


1  -f  W  =  0 ; 


{*) 


CONDITIONS  OF  EQUILIBRIUM. 


43 


iction   as  an 

force. 

and  «,  /J,  y, 

;e  axes  of  ;/•, 

sum  of  tlie 
irallel  to  the 
)e  considc'ivd 
)8  the  parti- 
rallcl  to  the 


(1) 


if  the  given 

\  iirticle   to 

iidl.  Geom., 


in  which 


(3) 


?r  which  tlie 
te  ])lanes  vz 
rnml  is  pcr- 
)  at  (.r,  //,  z). 
alar  to  fhc 
1.,    Art.    a?. 

(8) 
(4) 


ax        ,       dx'      ,_  dy      tj  _d£ 


dx 
dz' 


(Calculus,  Art.  56«.)     Substituting  in  (3)  and  (4),  we  have 

dx     dx'  _ 

^  +  d-z'd7-^* 
.,dydy[_Q, 


and 

from  which 


du 

^  =  -  ^  =  4^  (.Cal.  Art.  87)  =  a',  (5) 

dz'  dx        du 

dz 

du 
.         dy'  dz  _Ty_  ,,  (6) 

dz 
Substituting  these  values  of  a'  and  b'  in  (2)  and  multiply- 
ing both  terms  of  the  fraction  by  ^-,  we  have 


du 
dx 


cos  a  = 


cos  (i  = 


V(I)'-(|F-(S)' 


du 

dy 


duV' , 


^/m-a-'^ 


du 

dz 


cos  y  =  — 


\/o'+ cp^  cr 


(7) 


rtMta 


44 


COiWITIONS    OF   KQUILIBRWM. 


which  give  the  values  of  the  direction  cosines  of  the  normal 
at  (.r,  y,  z). 

Putting  the  denominator  equal  to  Q,  for  shortness,  and 
substituting  in  (1)  and  transposing,  we  have 


(8) 


du 
di' 

^          Q 

du 

z-     ^ 

du 
di 

X 

Y 

Z 

du  ~ 

du 

~  du' 

dx 

dy 

dz 

(9) 
(10) 


Eliminating  N  between  these  three  equations,  we  obtain 
the  two  independent  equations, 


(11) 


which  express  the  conditions  that  must  exist  among  the 
applied  forces  and  their  directions  in  order  that  their 
resultant  may  be  normal  to  tiio  surface,  i.  e.,  that  there  may 
be  equilibriuni.  If  these  two  equations  are  not  satisfied, 
equilibrium  on  the  surface  cannot  uxist.  Hence  the  point 
on  a  given  surface,  at  wliich  a  given  particle  under  the 
action  of  given  forces  will  rest  in  equilibrium,  is  the  point 
at  which  equations  (11)  arc  satisfied. 

CoK.  1. — Squaring  equations  (8),  (!)),  (10)  and  adding,  wc 
get 

'(du\»      (duV>       /duy 
W        V/y/^   ,    \dz)_ 

\_'Q'   +    <?»   ^    g> 


A'3  +  r«  +  z»  =  N^ 


=  N^; 


a  =  Vx»  +  r»  +  z», 


(12) 


of  the  normul 
shortness,  and 

(8) 

(0) 

(10) 
ns,  we  obtain 

(11) 


t  among  the 
r  that  their 
lat  there  may 
not  satisfied, 
ce  the  point 
le  under  tlie 
is  the  point 


d  adding,  we 

=  N^; 
(18) 


EXAMPLES. 


45 


which  is  the  value  of  the  normal  resistance  of  the  surface 
and  is  precisely  the  same  as  the  resultant  of  the  acting 
forces,  as  it  clearly  should  be  ;  but  this  resistance  must  act 
in  the  direction  opposite  to  that  of  the  resultant. 

Cor.  2. — Multiplying  (8),  (9),  (10)  by  dx,  dy,  dz,  respec- 
tively, and  adding,  and  remembering  that  the  total  differ- 
ential of  w  =  0  is  zero,  we  get 


Xdx  +  Ydy  +  Zdz  =  0, 


(13) 


which  is  an  equation  of  condition  for  equilibrium.  If  (13) 
cannot  be  satisfied  at  any  point  of  the  surface,  equilibrium 
is  impossible. 

Cor.  3. — If  the  forces  all  act  in  one  plane,  the  surface 
becomes  a  plane  curve ;  lot  this  curve  be  in  the  plane  xy, 
then  z  —  Q;  therefore  (11)  and  (13)  become 


(14) 


X 

Y 

du 

-  du' 

dx 

dy 

and 


Xdx  +  Ydy  =  0, 


(15) 


in  which  (14)  or  (15)  may  be  used  according  as  the  equation 
of  the  curve  is  given  as  an  implicit  or  explicit  function. 

EXAMPLES. 

1.  A  particle  is  placed  on  the  surface  of  an  ellipsoid,  and 
is  acted  on  by  attracting  forces  which  vary  directly  as  the 
distance  of  the  particle  from  the  princii)al  planes*  of  sec- 
tion ;  it  is  recjuired  to  determine  the  position  of  equilibrium. 

Let  the  equation  of  the  ellipsoid  be 


M  =  ./■  (x,  y, ») 


.2       ,,a       ^a 


*  Plauos  of  xy,  y*,  tx. 


itaii 


46 


EXAMPLES. 


du  _  2x      du  _  2y      du  _  2z 
•'•    dx~¥'     Ty"!^'    dz~J'' 

Aiid  let  the  x-,  y-,    and  z-components   of  the  forces  be 
respectively, 

X=— Hi*,    Y  =  —u^y,  Z  =  —u^&; 
then  (11)  will  give 

which  may  be  put  in  the  form 


Ml 


«2     _   «3 


6-2 


If  those  conditions  arc  fulfilled,  the  particle  will  rest  at  all 
points  of  the  surface. 

2.  Again,  take  the  same  surface,  and  let  the  forces  vary 
inversely  as  the  distances  of  the  point  from  the  principal 
planes;  it  is  required  to  determine  the  position  of  equili- 
brium. 


Here     X=:--',     r=z-^^,     Z  = 
x'  y  ' 


u. 


therefore  (11)  becomes 

^        «/2         «« 

fl»  _  J^    _  ^  -_ ^-  

Ui    ~  «sj   "~  V_^   ~  U^   +  Mjj  +  t«8 

by  putting  u  for  Hy  +  u^  +  ti.^. 


1 

u 


iun 


■<-\ 


EXAMPLES. 


forces  be 


rest  at  all 


forces  vary 
:;  principal 
L  of  equili- 


which  in  (12)  gives 


1  «2 


u. 


x^ 


y* 


- ''  L«^  "^  li'  +  c^J* 


47 


3.  A  particle  is  placed  inside  a  smooth  sphere  on  the  con- 
cave surface,  and  is  acted  on  by  gravity  and  by  a  repulsive 
force  which  varies  inversely  as  the  square  of  the  distance 
from  the  lowest  point  of  the  sphere;  find  the  position  of 
equilibrium  of  the  particle. 

Let  the  lowest  point  of  the  sphere  be  taken  for  the  origin 
of  co-ordinates,  and  let  the  axis  of  z  be  vertical,  and  posi- 
tive upwards;  then  the  equation  of  the  s])here,  whose 
radius  is  a,  is 

a;'  +  ^2  +  «^  -  2(72  =  0. 

Let  IF  =  the  weight  of  the  particle,  and  r  =  the  distance 
of  it  from  the  lowest  point;  then 


r*  =  .t2  +  %f  ^z^  ~  2az. 

Also,  let  the  repulsive  force  at  the  unit's  distaace 
then  at  the  distance  r  it  will  be 


ui 


u 


u 
2az* 


.*.    X  = 


2az 


(S 


')'. 


r  = 


z  = 


u 
Uz 


u 

202 


?/ 


-  W. 


48 


EXAMPLES. 


Let  N  =  the  normal  pressure  of  the  curve  ;  then  (8)  and 
(10)  give 

2az     r  a 


2az     r  a 


from  which  we  have 


r3  = 


na 


z  = 


u^ 


2a^W^' 


whence  the  position  of  the  particle  is  known  for  a  given 
weight,  and  for  a  given  value  of  n.  (See  Price's  Anal. 
Mechanics,  Vol.  I,  p.  39.) 

4.  Two  weights,  P  and  Q,  are  fastened  to  the  ends  of  a 
string,  (Fig.  11),  which  passes  over  a  pulley,  0 ;  and  Q 
hangs  treeij'  when  P  rests  on  a  plane  curve,  AP,  in  a 
vertical  plane  ;  it  is  required  to  find  the  position  of  equili- 
brium when  the  curve  is  given. 

The  forces  which  act  on  P  are  (1)  the 
tension  of  the  string  in  the  line  OP,  which 
is  equal  to  the  weight  of  Q,  (2)  the  weight 
of  P  acting  vertically  downwards,  (3)  the 
normal  reaction  of  the  curve  P. 

Let  0  bo  the  origin  of  co-ordinates,  and 
the  axis  of  x  vertical  and  positive  down- 
wards. Let  OM  ^  X,  MP  =  y,  OP  =  r, 
FOM  =  e,  OA  rzz  a.     Then, 


X  =  P  —  Q  cos  0—  R 


dy 


Fig.  II 


Y=  -Qmxd  +  R^^; 


hen  (8)  am 


for  a  given 
'rice's  Anal. 


he  ends  of  a 
,  0 ;  anil  Q 
e,  AP,  in  a 
on  of  equili- 


Fig.ll 


EXAMPLES. 

therefore  from  (15)  we  have 

{P  —  Q  cos  6)  dx  —  Q  sin  ddy  =  0, 


or 

But  since 
we  have 


X  dx  -{-  y  (ty 

Pdx  —  Q L_e^-j?  —  0. 

r 


a?  +  y^  =  r', 
xdx  +  y  dy  =  rdr  ; 
.'.    Pdx—Qdr  =  0', 


40 


(1) 


which  IS  the  condition  that  must  he  satisfied  by  J',  Q,  and 
the  equation  of  the  curve. 

5.  Required  the  equation  of  the  curve,  on  all  points  of 
which  P  will  rest. 


Integrating  (1)  of  Ex.  4,  we  have 

Px  —  Qr  =  a 


(1) 


But  since  P  is  to  rest  at  all  points  of  the  curve,  this  equa- 
tion must  be  satisfied  when  /'  is  at  A,  from  which  we  get 
X  =  r  =  a',  therefore  (I)  becomes 


which  in  (1)  gives 


Pa-  Qa=  C; 


.z. • 

P  ' 

1  —  -^  cos  0 


r  = 


which  is  the  equation  of  a  coni(^  section,  of  which  the  focus 
is  at  the  pole  0  ;  and  is  an  ellipse,  parabola,  or  hyperljola, 
according  as  P  <,  =,  or  >  Q. 
3 


^ 


50 


EXA  MI'LES. 


EXAMPLES, 


1.  Two  forces  of  10  and  :iO  lbs.  act  on  a  particle  at  an 


angli'  of  CO' ;  Hnd  the  resultant 


Alls.  :iG.5  lbs. 


' 'i.  The  resultant  of  two  forces  is  10  lbs.;  one  of  tlic 
forces  is  8  lbs.,  and  the  other  is  inclined  to  the  resultant  at 
an  an^^le  of  ;]6°.  Find  it,  and  also  find  the  angle  between 
the  two  forces.  (There  are  two  solutions,  this  being  the 
ambiguous  ease  in  the  solution  of  a  triangle.) 

Ann.  Force  is  :;>.6G  lbs.,  or  13.52  lbs.  Angle  is  47°  17' 
05',  or  i;J2°  43'  55". 

'  3.  A  point  is  kept  at  rest  by  forces  of  0,  8,  11  lbs. 
Find  the  angle  between  the  forces  0  and  8. 

Ann.   77°  21' 53". 

'4.  The  directions  of  two  forces  acting  at  a  point  are 
inclined  to  each  other  (1)  at  an  angle  of  (J0%  (2)  at  an 
angle  ot^  120",  and  the  resi)ective  resultants  are  as 
Vl  :  V3  ;  compare  the  magnitude  of  the  forces. 

Ans.  2  :  1. 

5.  Three  posts  are  placed  in  the  ground  so  as  to  form  an 
equilateral  triangle,  and  an  elastic  string  is  stretched  round 
them,  the  tension  of  which  is  0  lbs. ;  find  the  pressure  on 

■  G.  The  angle  between  two  unknown  forces  is  37°,  and 
their  resultant  div'  'es  this  angle  into  31  and  G" ;  find  the 
ratio  of  the  component  forces.  .|«.y.  4.!)27  :  1. 

7.  If  two  equal  rafters  sup])ort  a  weight,  W,  at  their 
upi)er  ends,  required  the  com])ression  on  each.  Let  the 
length  of  each  rafter  l)c  a,  and  the  horizontal  distance 
between  their  lower  ends  be  b.  .  a  W 


Ans. 


V4a2  -b^ 


KXAMPTjES. 


51 


(article  at  an 
.-.  -iG.b  lbs. 

one  of  tlu' 

resultant  at 

iglo  between 

Is  being  tlie 

gle  is  47°  17' 


G,  8,  11  lbs. 

7°  21'  53". 

a  point  are 
J",  (2)  at  an 
Mits    are    as 
es. 
us.  2  :  1. 

s  to  form  an 
tchcd  round 
pressure  on 
ns.  G  \/y. 

is  37°,  and 
5" ;  find  the 
4.!)27  :  1. 

ir,  at  tlieir 
h.  Let  tlio 
tal  distanec 

ffir 


8.  Three  forces  act  at  a  point,  and  include  angles  of 

00    !uid  45 '.    The  first  two  forces  are  each  equal  to  iP, 

and  the  resultant  of  them  all  is  VlOP;  find  the  third 
'■"'•t-'C-  Ans.  r  y/-i. 

'9.  Find  the  magnitude,  R,  and  direction,  B,  of  the 
resultant  of  the  three  forces.  /',  =  30  lbs.,  I\  =  70  lbs., 
7*3  =  50  lbs.,  tiie  angle  included  l)etween  J\  and  P., 
being  5G°,  and  between  P^  and  P^  104°.  (It  is  generally 
convenient  to  take  the  action  line  of  one  of  the  forces  fur 
the  axis  of  x. ) 

Let  the  axis  of  x  coincide  with  the  direction  of  P^;  then 
(Art.  3G),  we  have 

X  =  22.1G  ;     r  =  75.13  ;     R  =  78.33  ;     0  rr  73°  34'. 

10.  Tiirec  forces  of  10  lbs.  eacii  act  at  the  same  point ; 
the  second  makes  an  angle  of  30 '  with  the  first,  and  the 
third  makes  an  angle  of  G0°  with  the  second  ;  find  the 
magnitude  of  the  resultant.  J  ha-.  24  lbs.,  nearly, 

-^  11.  If  three  forces  of  t)9,  100,  and  101  units  respectively, 
act  on  a  point  at  angles  of  120";  find  the  magnitude  of 
their  resultant,  and  its  inclination  to  the  force  of  100. 

J«.v.   V3;  90^ 

12.  A  block  of  800  lbs.  is  so  situated  that  it  receives 
from  the  Avater  a  pressure  of  400  lbs.  in  a  south  direction, 
and  a  pressure  from  tiie  wind  of  100  lbs.  in  a  westerly 
direction  ;  required  the  niagnitude  of  the  resultant  j)res- 
sure,  and  its  direction  with  the  vertical. 

Ans.   900  lbs. ;  27°  IG'. 

^  13.  A  weigliT  of  40  lbs.  is  supported  by  two  strinps.  one 
of  wiiicii  makes  an  angle  of  30°  witii  the  vertical,  the  other 
ly ;  find  the  tension  in  each  string. 

Aiis.  20  (Vg  -  V2) ;  40  (a/3  —  1). 


52 


EXAMPLES. 


14.  Two  forces,  P  and  /",  acting  along  the  diagonals  of 
a  parallelogram,  keep  it  at  rest  in  such  a  podition  that  one 
of  its  sides  is  horizontal;  show  that 

P  sec  «'  =  P'  sec  n  =  IV  cosec  («  +  a'), 

where  U'  is  the  weight  of  the  parallelogram,  and  «  and  «' 
the  angles  between  the  diagonals  and  the  horizontal  side. 

15.  Two  persons  pull  a  heavy  Avcight  by  ropes  inclineil 
to  the  horizon  at  angles  of  G0°  and  30°  with  forces  of 
160  Ibfi.  and  200  lbs.  Tlie  angle  between  the  two  vertical 
planes  of  tlie  ropes  is  30°  ;  find  the  single  horizontal  force 
tliat  would  products  the  same  effect.  Atis.  245.8  lbs. 

IG.  In  order  to  raise  vertically  a  heavy  weight  by  means 
of  a  rope  passing  over  a  fixed  pulley,  three  workmen  pull  at 
the  end  of  the  rope  with  forces  of  40  lbs.,  50  lbs.,  ami 
100  lbs. ;  the  directions  of  these  forces  being  inclined  to 
the  horizon  at  an  angle  of  00°.  What  is  the  magnitude  ol 
the  resultant  force  which  tends  directly  to  raise  the  weight? 

A  us.  1(14.54  11)8. 

17.  Three  persons  pnll  a  heavy  weight  by  cords  inclined 
to  the  hori'^on  at  an  angle  of  GO",  with  forces  of  100,  120, 

,  and  140  lbs.  The  three  vertical  planes  of  the  cords  arc 
inclined  to  each  other  at  angles  of  30° ;  find  the  single 
horizontal  force  that  would  produce  the  same  effect. 

Aus.  10  V'US  +  72  V3  lbs. 

18.  Two  forces,  ^  aiul  Q,  acting  respectively  parallel  to 
the  base  and  length  of  an  inclined  plane,  will  each  singls 
sustain  on  it  a  particle  of  weight,  )(';  to  determine  tln' 
weight  of  ir. 

Fict  «•  =  incliiuitioii  of  (lie  plane  to  (lie  horizon;  then 
resniving  in  each  case  along  the  jilane,  so  I  hat  the  normal 
pressures  nuiy  not  enter  into  the  etjuutions  (See  Kern.,  Ex.  3, 
Art.  41)i  wo  hjivy 


£X  AM  PLUS. 


53 


diagonals  of 
tior  that  one 


ind  a  and  «' 
zontal  side. 

opes  inclineil 
nth  forces  of 
)  two  vertical 
rizontal  force 
'.  245.8  lbs. 

gilt  by  means 
■kmcn  pull  at 
,  50  lbs.,  and 
g  inclined  to 
magnitude  ol 
e  the  weight? 
104.54  lbs. 

ords  inclined 
of  100,  120, 
the  cords  are 
id  the  single 
effect. 

73  V3  lbs. 

ly  parallel  td 
1  each  singly 
etcrmino  the 


lorizon ;  then 
it  the  normal 
3  Kom.,  Kx.  13, 


P  cos  «  =  W  sin  «  :     Q  =  W  sin  « ; 


W 


19.  A  cord  whose  length  is  21,  is  fa.stencd  at  A  and  B,  in 
the  same  liorizontal  line,  at  a  distance  from  each  otiier 
equal  to  2a ;  and  a  smootli  ring  upon  the  cord  sustains  a 
wcigiit  11';  find  the  tension  of  the  cord. 

Ans.   T 


2  V?^  -  «* 

20.  A  heavy  particle,  whose  weight  is  W,  is  sustained  on 
a  smootli  inclined  plane  by  three  forces  applied  to  it,  each 

W 

equal  to  — ;  one  acts  vertically  upward,  another  horizon- 
o 

tally,  and  the  third  along  the  plane  ;   find  the  inclination, 
«,  of  the  plane. 


Ans.  tan  -  =  -• 

/&  At 


"21.  A  body  whose  weight  is  10  ^bs.  is  supported  on  a 
smooth  inclined  plane  by  a  force  of  2  lbs.  acting  along  the 
plane,  and  a  horizontal  force  of  5  lbs.  Find  the  inclination 
of  the  plane.  Ans.  siu~'  |. 

22.  A  body  is  sustained  on  a  smooth  inclined  plane  (in- 
clination «)  by  a  force,  P,  acting  along  the  plane,  and  a 
horizontal  force,  Q.  When  the  inclination  is  halved,  and 
the  forces,  P  and  Q,  each  halved,  the  body  is  still  observed 


to  rcoL ;  find  the  ratio  of  P  to  Q. 


P 
Ans.  jj  =  2  cos^ 


4 


23.  Two  weights,  P  and  Q,  (Fig.  12),  rest 
on  a  smooth  double-inclined  plane,  and  are 
attached  to  the  extremities  of  a  string 
which  passes  over  a  smooth  ])eg,  0,  at  a 
point  vertically  over  th(!  intersection  of  the 
planes,  the  peg  and  tiie  weights 


Fl9.« 


ili 


^^ 


54 


t;XAMPLlSS. 


vertical  rlane.     Find  the  position  of  equilibrium,  it  I  —  the 
length  oi  the  string  and  h  =  CO. 

Ans.  The  position  of  ciiuiiibrium  is  given  by  the  equa- 
tions 

_  sin  tc  _  ^  ^}P_^ 
cos  (^  ~      cos  f ' 

cos  «      cos  0  _  } 
sin  0       sin  (j)  ~  h 

■'  24.  Two  weights,  P  and  Q,  connected  by  a  string, 
length  I,  rest  on  the  convex  side  of  a  smooth  vertical 
circle,  radius  a.  Find  the  position  of  equilibrium,  and 
show  tiiat  the  iieavier  weight  will  be  higher  up  on  the 
circle  than  the  lighter,  the  radius  of  the  circle  drawn  to  P 
making  an  angle  0  with  the  vertical  iliameter. 

Ans.  P  sin  0  ~  ^  sin  I    —  dj- 

"125.  Two  weights,  P  and  Q,  connected  directly  by  a 
string  of  given  length,  rest  on  the  convex  side  of  a  smooth 
vertical  circle,  the  string  forming  a  chord  of  the  circle  ; 
find  the  position  of  equilibrium. 

Arts.  IF  2rt  is  the  angle  .subtended  at  the  centre  of  the 
circle  by  the  string,  the  inclination,  0,  of  the  string  to  the 
vertical  is  given  by  the  equation 

P—0 

cot  0  =  p  77^  tan  «. 


26.  Two  weights,  P  and  Q,  (Fig.  13), 
rest  on  the  concave  siJe  of  a  parabola 
whose  axis  is  horizontal,  and  are  con- 
nected l)y  a  string,  length  /,  which 
pa.sses  over  a  smooth  peg  at  the  focus,  /'. 
Find  (he  position  of  equilii)rium. 

Ans.  Let   0  =  the  angle   which   FP 


Fig.13        <j 


1,  it  J  =;  the 
jy  the  cqua- 


)y  a  string, 
)oth  vertical 
ibrium,  and 
r  up  on  the 
drawn  to  P 

irectly   by  a 
of  a  smooth 
the  circle  ; 

ipntre  of  the 
string  to  the 


Flg.13        u 


rXAMPLBS. 


55 


makes  with  the  axis,  and  4>«  =  the  latiis  rectum  of  tlie 
parabola,  then 


cot 


^       Vw  (/^  +  Q^) 


27.  A  jiarticle  is  placed  on  the  convex  side  of  a  smooth 
jUipse,  and  is  acted  upon  by  two  forces,  F  and  F',  towards 
the  foci,  and  a  force,  F'',  towards  the  centre.  Find  the 
position  of  equilibrium. 

Ans.  r  =  — ,  where  r  ■=  the  distance  of  the  par- 

Vl  -  n^ 
tide  from  the  centre  of  the  ellipse ;  b  =  semi-minor  axis, 

F-F' 

and  n  =  —yr- 

28.  TiCt  the  curve,  (Fig.  11),  be  a  circle  in  Avhich  the 
origin  and  pulley  are  at  a  distance,  a,  above  the  centre  of 
the  circle  ;  to  determine  the  position  of  equilibrium. 

Q 

Ans.  r  =  -p  a. 

29.  lict  the  curve,  (Fig.  11),  bo  a  hyperbola  in  which  the 
origin  and  i)ulley  are  at  the  centre,  0,  the  transverse  axis 
being  vertical  ;  to  determine  the  position  of  e(|uilil)rium. 

Aiis,  X  = T- 

30.  A  particle,  P.  is  acted  upon  by  two  forces  towards 
two  fixed  points,  S  and  H,  these  forces  being  -^,:  and   .jj,, 

Of  111 

respectively;  prove  that  P  will  rest  at  all  points  inside  a 
!<mooth  tube  in  the  form  of  a  curve  whose  equation  is  SP. 
PH  =  P,  k  being  a  (ionstant. 

31.  Two  weights,  /'  and  Q,  connected  by  a  string,  rest 
on  the  convex  side  of  a  smooth  cycloid.  Find  the  position 
of  equilibrium. 


50 


EXAMPLES. 


Ans.  If  I  =  the  length  of  the  string,  and  a  =  radius  of 

generating  oircle,  the  position  of  e(iuilibrium  ii  defined  by 

the  equation 

•     «  _        ^  J 

'"'  2  -  P  +  Q  '  4«' 

where  0  is  the  angle  between  the  vertical  and  the  radius  to 
the  point  on  the  generating  circle  which  corresponds  to  J'. 

32.  Two  weights,  /'  and  Q,  rest  on  the  convex  side  of  a 
smooth  vertical  circle,  and  are  connected  by  a  string  whicli 
pusses  over  a  smooth  peg  vertically  over  the  centre  of  the 
circle  ;  find  the  position  of  equilibrium. 

Ans.  Let  h  —  the  distance  between  the  peg,  B,  and  the 
centre  of  the  circle  ;  0  and  </>  =  the  angles  made  with  the 
vertical  by  the  radii  to  P  and  Q,  respectively  ;  «  and  /J  — 
the  angles  made  with  the  tangents  to  the  circle  at  J'  and 
Q  by  the  portions  PB  and  QB  of  the  string;  I  =■  lengtli 
of  the  string;  then 

psin^  _  /)""  "^ 
cos  rt  ~~  ^  cos  i3 ' 


,  /sin  0       sin  0\        , 
Vcos  a       cos  (3/ 

h  cos  {6  +  «)  =  rt  cos  a, 

h  cos  {(p  +  (i)  =  a  cos  /3. 


=  radius  of 
<  defined  by 


the  radius  to 
sponds  to  J'. 

,ex  side  of  a 
string  whicli 
;entre  of  the 

f,  B,  and  the 
lado  with  tlie 
rt  and  i3  = 
•clo  at  J'  and 
;  I  =.  Icngtli 


CHAPTER     III. 

COMPOSITION    AND    RESOLUTION    OF   FORCES  ACTING 
ON    A    RIGID    BODY. 

43.  A  Rigid  Body.— In  the  last  cluipter  we  considered 
the  action  of  forces  whicli  have  a  com i  sun  point  of  applica- 
tion. We  shall  now  consider  the  action  of  forces  which  are 
applied  at  diiferent  i)oi»its  of  a  rigid  body. 

A  rigid  body  is  one  in  which  the  particles  retain  invari- 
able positions  with  respect  to  one  another,  so  that  no 
external  force  can  alter  tlieni.  Now,  as  a  matter  of  fact, 
there  is  no  such  thing  in  nature  as  a  body  that  is  perfectly 
rigid  ;  every  body  yields  more  or  le^s  to  the  forces  which 
act  on  it.  If,  then,  in  any  case,  the  body  is  altered  or  com- 
pressed aj)prociably,  we  shall  suppose  that  it  has  assumed 
its  figure  of  equilibrium,  and  then  consider  the  points  of 
application  of  the  forces  as  a  system  of  invarial)le  form. 
The  term  body  in  this  work  means  rigid  body. 

44.  Transmissibility  of  Force.— When  a  force  acts 
at  a  definite  point  of  a  I'ody  and  along  a  definite  line,  the 
effect  of  the  force  will  be  unchanged  at  whatever  point  of 
>ts  direction  we  suppose  it  ai)|)lied,  ])rovidcd  this  point  be 
either  one  of  the  points  of  the  body,  or  l)e  invariably  con- 
uected  with  the  body.  This  jirinciple  is  called  .the  tratis- 
viissibility  of  a  force  to  any  point  in  its  line  of  action. 

Now  two  e(]ual  forces  acting  on  a  i)article  in  the  same 
line  and  in  opposite  directions  neutralize  each  other  (Art. 
16)  ;  so  l)y  this  prmciph^  two  equal  forces  acting  in  the 
«amo  lino  and  in  opjtosite  directions  at  any  points  of  a 
rigid  body  in  that  line  neutralize  each  other.  Hence  it  is 
<'lear  that  when  many  forces  are  acting  on  a  rigid  body, 
any  two,  which  are  equal  and  have  the  same  line  of  action 


^M 


58 


RESULTANT  OF  PARALLEL  FORCES. 


and  act  in  oi)po8itc  directions,  may  be  omitted,  and  alsn 
that  two  equal  forces  along  llie  same  line  of  action  and  in 
opposite  directioi's,  may  ho  inlroduced  without  changing 
tlij  tircumstances  of  the  system. 


45.  Resultant  of  Two  Parallel 

Forces.*— (1)  Let  P  and  Q,  (Fig. 
14),  be  tiie  two  parallel  forces  acting 
ot  the  point'i  A  and  B,  in  the  same 
direction,  on  a  rigid  body.  It  is  re- 
quired to  find  the  resultant  of  P 
and  Q. 

At  A  and  B  introduce  two  equal 
and  opposite  forces,  F.  The  introduction  of  these  forces 
will  not  disturb  the  action  of  /'  and  Q  (Art.  44).  Pand  F 
at  A  are  equivalent  to  a  single  forco,  R,  aad  Q  and  F  at  M 
are  equivalent  to  a  single  force,  S.  Then  let  R  and  8  bo 
supposed  to  act  at  0,  the  point  of  in;ersection  of  their  lines 
of  action.  At  this  point  let  them  be  resolved  into  their 
components,  P,  F,  and  Q,  F,  respectively.  The  two  forces, 
F,  at  0,  neutralize  each  other,  while  the  components,  P 
and  Q,  act  in  the  line  OG,  parallel  to  their  lines  of  action 
at  A  and  B.  Hence  the  mngnitnde  of  the  resultant  is 
P+  Q,  (Art.  28).  To  find  the  point,  G,  in  which  its  lino 
of  aciion  cqts  AB,  let  the  extremities  of  Pand  R  (acting  at 
A)  be  joined,  and  complete  the  parallelogram.  Then  the 
triangle  PAR  is  evidently  similar  to  GOA  ;  therefore, 

P       GO      ....    (?       GO 
^=:  g-^;  similarly  ^=(j3i 


therefore,  by  division, 


P 

Q 


GB 
GA" 


(1) 


*  Mltichln'H  Statics,  p.  86. 


s. 

cd,  and  also 
r'tion  and  in 
lit  changing 


these  forces 
).  Pand/'' 
and  F  at  }^ 
E  and  S  bo 
of  their  lines 
id  into  their 
le  two  forces, 
nponents,  P 
les  of  action 

resultant  is 
hich  its  line 

Ji  (acting  at 
.  Then  the 
refore, 


(1) 


RBStlLfANT  OF  PARALLEL  FORCES. 


h\i 


P-Q 


(2)  When  ike  forces  act  in  opposite  directions. — At  A  and 
B,  (Fig  15),  apply  two  equal  and  opposite  forces  /•',  as 
before,  and  let  A',  the  resultant  of  P 
and  F,  and  <S',  the  resultant  of  Q  and 
F,  be  transferred  to  0,  their  point  of 
intersection.  If  at  0  the  forces.  It 
and    *S',  are    decomposed    into    their  _ 

original  components,  tiie  two  forces,  f    o    f 

F,  destroy  each  other,  the  force,  P,  ^'^'^ 

will  act  in  the  direction  GO  parallel  to  the  direction  of 
P  and  Q,  and  the  force  Q  will  act  in  the  direction  OG. 
Hence  the  resultant  is  a  force  =z  P  —  Q,  acting  in  the  line 
GO.  To  find  the  point  G,  we  have,  from  the  similar 
triangles,  PAR  and  OGA, 


P_QO     ,     Q 
F-GA'  ""''"  F 


GO 
GB ' 


P 
Q 


GB 
GA' 


(2) 


Hence  the  resultant  of  ttoo  parallel  forces,  actiiig  in  the 
same  or  opposite  directions,  at  the  extremities  of  a  rigid 
right  line,  is  parallel  to  the  components^  equal  to  their 
alffebrnic  sum,  and  divides  the  line  or  the  line  produced, 
into  two  segments  which  are  inversely  as  the  forces. 

In  both  cases  we  have  the  equation 


P  X  GA  =  ^  X  GB. 
Hence  the  following  theorem  : 


(3) 


If  from  a  point  on  the  resultant  of  two  parallel  forces  a 
right  line  be  drawn  meeting  the  forces,  whether  perpendicu- 
larly or  not,  the  products  obtained  by  vniUiplying  each  force 
by  its  distance  from  the  resultant,  measured  along  the  arbi- 
trary line,  arc  equal. 

ScH. — The  point  G  possesses  this  remarkable  property ; 


^^ 


CO 


MOMKXT    OF   A    FORCE. 


that,  however  P  and  Q  are  turned  about  their  poii  i  of 
application,  A  and  B,  tlieir  directions  ifp  linin'?  pavullel, 
G,  deteimined  as  allow  '•niaiiiri  iixi-d.  Tins  point  is  \n 
conse^juenoe  caied  the  centre  of  the  parallel  forces,  P 
and  Q. 

46.  Moment  of  a  Force. —  The  moment  of  n  fo)ce  with 
respect  to  (i  jwint  is  t/ie  product  of  the  force  and  tl>.e  perpen- 
dicular let  fall  OH  its  line  of  action  from  the  point.  The 
moment  of  u  force  measures  its  tendency  to  produc  rota- 
tion ahout  a  fixed  point  or  fixed  axis. 
Thus  let  a  force,  P,  (Fig.  10),  act  on 
a  rigid  body  in  the  plane  of  the  paper, 
and  let  an  axis  perpendicular  to  this 
plane  pass  through  the  body  at  any 
point,  0.  It  is  clear  that  the  effect  of 
the  force  will  be  to  turn  the  body  round  this  axis  (the  axis 
being  supposed  to  be  fixed),  and  the  turning  effect  will 
depend  on  the  magnitude  of  the  force,  P,  and  the  perpen- 
dicular distance,  p,  of  /'  from  0.  If  P  passes  through  0, 
it  is  evident  that  no  rotation  of  the  body  round  0  can  take 
place,  whatever  be  the  magnitude  of  P\  while  if  P 
vanitnes,  no  rotation  will  take  place  however  great  p  may 
be.  ILmicc,  the  measure  of  the  power  of  the  force  to 
produce  rotation  may  be  represented  by  the  product 

P.  p, 

and  this  product  has  received  the  special  name  of  Moment. 

The  unit  of  force  being  a  pound  and  the  unit  of  length  a 
foot,  the  unit  of  moment  will  evidently  be  a  foot-pound. 

The  i)oint  0  is  csdled  the  origin  of  mommts,  and  may  or 
may  not  be  chosen  to  coincide  with  the  origin  of  co- 
ordinates. The  solution  of  proldems  is  often  greatly  sim- 
plified by  a  proper  s  'lection  of  the  origin  of  moments.  The 
perpendicular  from  the  origin  of  moments  to  the  action  line 
of  the  force  is  called  the  arm  of  (he  force. 


■n« 


SlOJfS    O:     MOMENTS. 


61 


pir  poirl    of 

\\y%  pavullel, 

point,  is  ill 

L'l    furcL'S,    P 


a  fo)  ce  ii'Uh 
I  the  perpen- 
point.  The 
roduc    rota- 


xis  (the  iixis 
g  effect   will 

I  tite  pcrpen- 
i  through  0, 

II  0  can  take 
while  if  P 
great  p  may 

the   force   to 
oduct 


of  Moment. 
it  of  length  a 
it-pouud. 
I,  and  may  or 
origin  of  co- 

grcatly  sim- 
oments.  The 
he  action  line 


47.  Signs  of  Moment**.— A  force  may  tend  to  tnrn  a 
body  about  a  j)oint  or  about  an  axis,  in  either  of  two  direc- 
tions; 1  one  be  regarded  as  positive  the  other  must  be 
nrgnlive ;  -v.J  I.-mcc  we  distinguish  between  positive  and 
ni'ijative  moments,  ^or  the  sake  of  uniformity  the  moment 
of  a  force  is  said  to  be  negative  \\\\Gn  it  tends  to  turn  a  body 
from  left  to  right,  /.  e.,  in  the  direction  in  which  the  hands 
of  a  clock  move  ;  and  positive  when  it  tends  to  turn  the 
body  from  right  to  left,  or  opposite  the  direction  in  whi- ', 
the  hands  of  a  clock  move. 

48.  Geometric  Representation  of  the  Mome<<  v 

a.  Force  with  respect  to  a  Point— Let  the  li      /.  K 

(Fig.  l(i),  represent  the  fc-'ie,  P,  in  magnitude  an<^,  ■\..'c. 
tion,  unAp  the  perpendicular  OC  ;  then  the  momei.'  ^f  /'' 
with  respect  to  0  is  ABx/'  (Art.  40).  But  this  it,  i'  >e 
the  area  of  the  triangle  AOB.  Hence,  tIte  moment  of  o,  force 
with  respect  to  a  point  is  (/eometrically  represented  by  double 
t/ie  area  of  the  triangle  whose  base  is  the  line  representing 
the  force  in  magnitude  and  direction,  and  whose  vertex  is 
the  given  point. 

49.  Case  of  Two  Equal  and  Opposite  Parallel 
Forces. — If  the  forces,  P  and  Q,  in  Art.  45,  (Fig.  15)  are 
equal,  the  equation 

P  X  GA  =  ()  X  GB 

gives  GA  =  GB,  which  is  true  only  when  G  is  at  infinity 
on  AB;  also  the  resultant,  P — Q,  is  equal  to  zero.  Such  a 
system  is  called  a  Conple. 

A  Couple  consists  of  two  eqval  and  tipposi(e  paralhl  forces 
acting  on  a  r'gid  bwtij  at  a  finite  disfanre  from  each  other. 

We  shall  investigate  the  laws  of  the  composition  iind 
resolution  of  couples,  since  to  these  the  composition  and 


•^  ■ 


(52 


MOMENT    OF   A    COUPLE. 


resolution  of  forces  of  every  kind  acting  on  a  rigid  body 
may  be  reduced. 


6     d 


50,  Moment  of  a  Couple.— Let  0     ♦? 

(Fig.  17)  be  any  point  in  the  plane  of  the 

couple ;   let  fall  the  perpendiculars  Oa 

and  Ob  on  the  action  lines  of  the  forces 

P.    Then  if  0  is  inside  the  lines  of  action 

of  the  forces,  both  forces  tend  to  produce  ^'8"" 

rotation  round  0  in  the  same  direction,  and  therefore  the 

eum  of  their  moments  is  equal  to 

P  (Oa  +  Ob),  or  P  xab 

If  the  point  chosen  is  0',  the  sum  of  the  moments  is 
evidently 

p  (O'rt  -  O'b),  or  P  X  ab, 

which  is  the  same  as  before.  lie  nee  the  ntioment  of  the 
couple  with  respect  to  all  points  in  its  plane  is  constant. 

The  Arm  of  a  couple  is  the  perpendicular  distance 
between  the  two  forces  of  the  couple. 

The  Moment  of  a  couple  is  the  product  of  the  arm  and 
ons  of  the  forces. 

The  Axis  of  a  couple  is  a  right  line  drawn  from  any 
chosen  point  perpendicular  to  the  jjUme  of  the  couple,  and 
of  such  length  as  to  represent  the  magnitude  of  the  mo- 
ment, and  in  such  direction  as  to  indicate  the  direction  in 
which  the  couple  tends  to  turn. 

As  the  motion,  in  Statics  is  only  virtunl,  and  not  actual, 
llio  direction  of  the  axis  is  fixed,  but  not  tho position  of  it; 
it  may  be  any  line  perpendicular  to  tlio  plane  of  the  couple, 
jind  may  bo  drawn  as  follows;  imagine  a  watcii  pliiced  in 
the  plane  in  which  several  couples  act.  Then  let  the  axes 
vf  those  couples  which  tend  to  produce  rotation   in   the 


rigid  body 


o      6     tf 

Fig. 17 

lerefore  the 


moments  is 


ncnt  of  the 
onstfint. 
ar  distance 

;he  arm  and 

1  from  any 
couple,  and 
i  of  the  mo- 
iircction  in 

not  actual, 
si/ id  11  of  it; 
'  the  couple, 
h  pliiced  in 
let  the  axes 
ion  in  the 


COUTLES. 


63 


direction  of  tiie  motion  of  the  hands  be  drawn  dowmvard 
through  the  back  of  the  watch,  and  tlieaxesof  those  which 
ti'ud  to  pioduco  the  coiitiary  rotation  be  drawn  itpward 
tbroHgii  tlie  face  of  the  watclj.  Thus  each  couple  is  com- 
pletely represented  by  its  axis,  which  is  drawn  upward  or 
downward  according  as  tiio  moment  of  the  couj)le  is  posi- 
tive or  negative ;  and  couplos  are  to  be  resolved  ami 
compounded  by  the  same  geometric  constructions  jjerformed 
with  reference  to  their  axes  as  forces  or  velocities,  with 
reference  to  the  lines  whicii  directly  represent  tliera. 

We  shall  now  give  three  propositions  showing  tliat  the 
effect  of  a  couple  is  not  altered  when  certain  changes  are 
made  with  respect  to  the  couple. 

51.  TJie  Effect  of  a  Couple  on  a  Rigid  Body  is  not 
altered  if  the  arm  be  turned  through  any  angle 
about  one  extremity  in  the  plane  of  the  Couple. 

Let  the  plane  of  the  paper  be  the 
plane  of  the  couple,  AB  the  arm  of 
the  original  couple,  AB'  its  new  posi- 
tion, and  P,  P,  the  forces.  At  A 
and  B'  respectively  introduce  two 
forces  each  e(pial  to  P,  with  their 
action  lines  perpendicular  to  the  arm 
AB',  and  opposite  in  direction  to 
each  other.  The  effect  of  the  given 
couple  is,  of  course,  unaltered  by  the  introduction  of  these 
forces.  Let  BAB'  —  20  ;  then  the  resultant  of  P  acting  at 
B,  and  of  P  acting  at  B',  whose  lines  of  action  meet  at  Q, 
is2Psin  6,  acting  along  the  bisector  AQ;  and  the  result- 
ant  of  P  acting  at  A  perpendicular  to  AB  and  of  P  per- 
pendicular to  AB',  is  ^>/'  sin  0,  acting  along  the  bisector 
AQ  in  a  direction  o])posite  to  the  former  resultant.  Hence 
these  two  resultiints  iicutralizo  each  other;  and  there 
remains  the  couple  whose  arm  is  AB',  and  whose  forces  are 
P,  P.    Hence  the  effect  of  the  couple  is  not  altered. 


64 


COUPLES. 


/ 


A^ 

\ 


jP 


-1: 


52.  Tlie  Effect  of  a  Couple  nn  a  Rl^id  Body  is 
not  altered  if  we  transfer  t/ie  Cnaple  to  any  oilier 
Parallel  Plane,  the  Ann  remaining  parallel  to 
itself. 

Let  AB  be  the  arm,  and  P,  P,  the 
forces  of  the  given  couple;  let  A'B' 
be  the  new  position  of  the  arm  par- 
allel to  AB.  At  A'  and  B'  apply  two 
equal  and  opposite  forces  each  equal 
to  P,  acting  perpendicular  to  A'B', 
and  in  a  plane  parallel  to  the  plane  of  ^'o-" 

the  original  couple.  'J'his  will  not  altev  the  effect  of  the 
given  couple.  Join  AB',  A'B,  bisecting  each  other  at  0  ; 
then  P  at  A  and  P  at  B',  acting  in  parallel  lines,  and  in 
the  same  direction,  are  efjuivalent  to  'IP  acting  at  0  ;  also 
P  at  B  and  P  at  A',  acting  in  parallel  lines  and  in  tiie 
same  direction,  are  equivalent  to  'iP  acting  at  O.  At  0 
therefore  these  two  resultants,  being  equal  and  opposite, 
neutralize  each  other ;  and  there  remains  the  couple  whose 
arm  is  A'B',  and  whose  forces  are  each  P,  ac*^ing  in  tiie 
same  directions  as  those  of  tiie  original  couple.  Hence  the 
effect  of  the  couple  is  not  altered. 

53.  Tlie  Effect  of  a  Couple  on  a  Rigid,  Body  is 
not  altered  if  we  replace  it  by  another  Couple  of 
jvhich  th-'.  Moment  is  the  same;  the  Plane  remain- 
ing the  i...  m,e  and  the  Arms  being  in  the  same 
straight  line  and  having  a 
common  extremity. 

Let  AB  be  the  arm,  and  P,  P,  the 
forces  of  the  given  couple,  and  sup- 
pose P  =  Q  +  P.  Produce  AB  to  C 
so  that 

AB   :   AC    ::     Q  :    P  {=  Q  +  R), 

AB  :   BC 


tP=Q+R 


Fig.20 


iP=Q+R 


and  therefore 


Q  :  R; 


(1) 
(2) 


FORCE    AAD    A     COUPLE. 


65 


/(I  Body  is 
(ttiij  other 
Parallel   to 


-,  At 

/ 

.0 

/ 

\ 

/  \ 

',p  \  P* 

^■1 

\i 


FiB.I9 

eflfect  of  tlio 

other  at  O ; 

lines,  and  in 

J  at  O  ;  also 

3  and  in  tlic 

at  O.    At  O 

lid  ojijiositc. 

.'Oil pie  whosi' 

-oMiig  in   the 

Uence  the 

id   Body  i.i 

•  Couple  of 

ie  remiiiii- 

the   same 


20 


(1) 

(2) 


at  C  introduce  opjjosite  forces  eacii  equal  to  Q  and  parallel 
to  P  ;  this  will  not  alter  the  effect  of  tlie  couple. 

Now  i2  at  A  and  Q  at  (J  will  balance  Q  ■{■  li  ixi  B  from 
(2)  and  (Art.  45);  hence  there  remain  the  forces,  Q,  Q, 
acting  on  the  arm,  AC,  wliicli  form  a  couple  whose  moment 
IS  equal  to  that  of  P,  P,  with  arm,  AB,  since  by  (1)  we 
have 

P  X  AB  =  g  X  AC. 
Hence  the  effect  of  the  couple  is  not  altered. 

Rem. — From  the  last  three  articles  it  appears  that  we 
may  change  a  couple  into  another  couj)le  of  equal  moment, 
and  transfer  it  to  any  position,  either  in  its  11  plane  or 
in  a  plane  parallel  to  its  own,  without  altering  the  effect  of 
the  coui)le.  Tlic  couple  must  remain  unchanged  so  far  as 
concerns  the  direction  oj  rotation  which  its  forces  would 
tend  to  give  the  arm,  i.  e.,  the  axis  of  tlic  couple  may  be 
removed  parallel  to  itself,  to  any  position  Avithin  the  body 
acted  on  by  the  couple,  while  the  direction  of  the  axis  from 
tiie  plane  of  the  couple  is  unaltered  (Art.  50). 

54.  A  Force  and,  a  Couple  acting  in  the  same 
Plane  on  a  Rigid  Body  are  equivalent  to  a,  Single 
Force. 

Let  the  force  be  F nn(\.  the  couple  {P,  n),  tiiat  is.  Pis 
the  magnitude  of  cacli  force  in  tlie  couple  whose  arm  is  a. 

Then  (Art.  53)  the  couple  (P,  a)  =  the  couple  (p,  ^-Y 

Let  this  latter  couple  be  moved  till  one  of  its  forces  acts  in 

till'  same  line  as  the  given  force,  F,  but  in  the  opposite 

diri- tion.     Tlie  given  force,  P,  will  then  be  destroyed,  and 
the  ■(■  will  remain  a  force,  P,  acting  in  the  same  direction 

as  the  given  one  and  at  a  iierpendicular  distance  from  it 

(iP 


66 


RESULTANT    OF    COUPLES. 


Cor. — A  force  and  a  couple  acting  on  a  rigid  hotly  cannot 
prodhcc  e(/niIibriHin.  A  coupln  can  Ije  i.'t  equilibrium  only 
with  an  equivalent  couple.  BquivaletU  couples  are  thoxe 
wliose  :nohients  are  equal.* 

The  resultant  of  several  cmiples  is  one  which  will  produce 
the  same  effect  singly  as  the  component  couples. 

55.  To  find  the  ResuUdnt  of  amj  number  nj 
Couples  acting  on  a  Body,  the  Planes  of  the 
Couples  being  parallel  to  each  other. 

Let  P,  Q,  R,  etc.,  be  the  forces,  iiiid  a,  h,  c,  etc.,  tlieir 
arms  rcspoctivel)'.  Suppose  all  ciie  couples  tninsferred  to 
the  same  plane  (Art.  53) ;  next,  let  them  all  be  transferred  so 
as  to  have  their  arms  in  the  sanie  straight  lino,  and  one 
extremity  common  (Art.  •'51)  ;  lastly,  let  them  be  replaced 
by  other  couples  having  the  same  arm  (Art.  53).  Ijet  «c  be 
the  common  arm,  and  Pj,  (?,,  i?j,  etc.,  the  new  forces, 
80  that 

Pj«  —  Pa,     Qitt  =  Qh,    Rytr  =  Re,  etc., 
then       P,  =  P-,    Q.  =  Q~,   R.  =  R-,  etc., 

i.e.,  the  new  forces  are  P-,  Q   ,  R-  ,  etc.,  acting  on  the 

common  arm  «.     Hence  their  resultant  will  be  a  couple  of 
which  each  force  equals 

pi  +  qI  +  „l  +  „tc., 

and  the  arm  =  «,  or  the  moment  equals 
Pa  +  Qlj  -\-  Re  -f  etc. 
If  DUO  of  the  coupleH.  as  Q,  act  in  a  direction  opposite  to 

*  The  moiuotiti)  ufcqulviileut  cuupluu  may  have  Uku  or  unllko  HigUH 


th 
ead 


■ 


ItESULTANT    OF    TWO    COUPLES. 


g: 


'0'?/  cannot 
^triuui  only 
nre  thone 

ill  produce 


iniber  nj 
s    of   Die 

etc.,  their 
isfeiTcd  to 
sisferred  ho 
0,  and  one 
0  replaced 

Ij«t  «  be 
ew  forces, 

:c.. 


^Z  on  the 
30uple  of 


)08ik'  to 

{UN 


the  otlier  couples  its  sign  will  be  negative,  and  the  force  at 
each  extremity  of  the  arm  of  the  resultant  couple  will  bo 

p'_'_  g*  +  /ei  +  etc. 
«  «  a 

Hence  the  moment  of  the  resultant  couple  is  equal  to  the 
algebraic  sum  of  the  moments  of  the  component  couples. 

56.   To   Find   the    Jlesioltant  of  tivo  Couples  not 
acting  in  the  same  I'lane.* 

Let  the  planes  of  tht  couples  be 
inclined  to  each  othci  at  an 
angle  y ;  let  the  couples  be  trans- 
ferred in  their  pianos  so  as  to 
have  the  same  arm  lying  along 
the  lino  of  intersection  of  tiio  two 
planes  ;  and  let  the  forces  of  the 

couples  thus  transferred  be  P  and  Q.  Lot  AB  be  the  com- 
mon uruK  Let  A*  be  the  resultant  of  the  forces  P  and  Q  at 
A  acting  in  the  direction  Ali  ;  and  of  P  and  Q  at  B  acting 
in  the  direction  lili.  Then  since  P  and  ^  at  A  are  parallel 
to  P  and  Q  at  1$  respectively,  thei'efore  II  at  A  is  parallel 
to  Ji  at  B.  Hence  tiie  two  couples  are  e([uivalent  to  the 
single  couple  li,  R,  acting  on  the  arm  AB  ;  and  since 
y'A^  =  y,  wo  have 


Ri  =  /'»  +  ga  +  2PQ  cos  y  (Art.  30). 


(1) 


Draw  Art,  Bi  perpendicular  to  the  jdanes  of  the  couples 
/',  /',  and  (>,  Q,  respectively,  and  proportional  in  length  to 
tiieir  niomentf. 

Draw  Av  perpendicular  to  the  plane  of  U,  li,  and  in  the 
same  proj)ortion  to  Art,  Wh,  lliat  the  moment  of  the  couple, 
/.'.  /.',  is  l(»  those  of  /'.  /'.  and  (>.  Q,  respectively.  Tiien 
Art,  \l),  Ar,  may  lie  taken  as  the  axes  of  /',  /' ;  Q,  Q\  and 

•  Todt.Hiitor'B  Stallcn,  j).  i-i.    AIno  Pmtt's  McclmiilcB,  p.  S6. 


rite 


C8 


RESVLTAST   OF    TWO    COUVLKS. 


li,  R,  rcspoctivoly  (Art.  50).  Now  tlie  three  straight  linen, 
Art,  Ar,  Kb,  make  the  same  angles  with  each  other  that 
A/',  A^,  A^  make  with  each  other;  also  they  are  iu  the 
same  proportion  iu  which 

AB  .  P,   AB  ■  /?,   AB  .  (2  are, 

or  in  which  P,  li,  Q  are. 

But  R  is  the  resultant  of  P  and  Q  ;  therefore  A<;  is  the 
diagonal  of  the  parallelogram  on  Aw,  A/>  (Art.  30). 

Hence  if  two  struujhf  lines,  having  a  common  exircvitfi/, 
represent  the  axes  of  tiro  couples,  that  diagonal  of  the 
parallelogram  described  on  these  straight  lines  as  adjacent 
sides  which  passes  through  their  common  extremity  repre- 
sents the  axis  of  the  resultant  couple. 

Cor, — Since  R  •  AB  is  the  axis  or  moment  of  the  result- 
ant couple,  wo  have  from  (1) 

ii?.AB'=  /'2.AB''+<?^AB'+2P.AB-(2-AB-cosy.  (!i) 

If  Ij  and  ;]/  represent  the  axes  or  moments  of  the  com- 
j)onent  couples  and  G,  that  of  the  resultant  couple,  (2) 
becomes 


(P  =  D  +  AP  +  2L  ■  M  cos  y. 


(3) 


Ren.  1. — If  A,  M,  N,  are  the  axes  of  three  comp'mont 
couples  which  act  in  ])l;nu's  at  right  angles  lo  one  aiioliier, 
and  G  the  axis  of  the  resultant  couj)le,  it  may  lasiiy  be 
shown  that 


0«  =  7^,2  -f  Tlfs  +  NK 


(•1) 


If  A..  (I,  V  1)0  tlie  angles  which   the  axis  of  the  resultant 
makes  with  tho'^e  of  the  compoueiits,  we  have 


L  M 

cos  A  --  -^ ,     cos  fl  =     ,  , 


cos  V 


N 
0 


traight  lines, 
li  other  tliiit 
y  iire  iu  tlie 


re  Ac  is  tlio 
30). 

H  ex/ rem  if  I/, 
'onal  of  the 
^  as  ndjdcenf 
cmity  reptc- 


)l  the  result- 

B-cosy.  (2) 

of  tile  com- 
couple,  (2) 

(3) 

comp'mont 
)iie  uiiolhor, 
ly  easily  l)i« 

(4) 

10  rosuItniK 

V 


VARWNOA'S    TIIEOKEM    OF  MOMEATS. 


G!J 


ScH.  2. — Ileuce,  conversely  any  couple  may  be  re])lacecl 
by  three  couples  acting  in  planes  at  rigiit  angles  to  one 
another;  their  moments  being  ^r  cos  A,,  (f  cos  fi,  6' cos  j' ; 
where  O  is  the  moment  of  the  given  couple,  and  A,  /t,  v  the 
angles  its  axis  makes  witli  the  axes  of  the  three  couples. 

Thus  the  composition  ami  resolution  of  couples  follow 
laws  similar  to  those  wliich  apply  to  forces,  the  axis  of  the 
couple  corresponding  to  the  direction  of  the  force,  and  the 
moment  of  the  conple  to  the  mafjiiitmle  of  the  force. 

57.   Varignon's  Theorem  of  Moments.— TAw  nw- 

rnetit  of  the  resultant  of  two  coiiiponent  forces 
with  respect  to  any  point  in  their  plane  is  ef/nal 
to  the  algehraie  sain,  of  the  moments  of  the  two 
components  with  respect  to  the  same  point. 

Let  A  P  and  A  Q  represent  two  com- 
ponent forces ;  oorn;)leto  the  parallelo- 
gram and  draw  tie  diagonal,  Ali, 
representing  the  resultant  force.  Let 
0  be  the  origin  of  mcments  (Art.  40). 
Join  OA,  OP,  OQ,  OR,  and  draw  PV 
and  QB  parallel  to  OA,  and  let  p  =  the  perpendicular  let 
fall  from  0  to  A  h\ 

Now  (he  moment  of  AP  about  0  is  the  product  of  AP 
and  the  perpendicular  lot  fall  on  it  from  0  (Art.  40),  which 
18  double  the  area  of  the  triangle,  A  OP  (Art.  48).  But 
the  area  of  the  triangle,  A  01',  =  the  area  of  the  triangle, 
A  00,  since  these  triangles  have  the  same  base,  AO,  and 
are  between  the  same  piirallels,  AO  and  CP.  Hence  tho 
moment  of  AP  about  0  ~  the  moment  of  AC  about 
O  --  AC -p.  Also  the  nnmienl  of  .(('about  0  is  double 
the  area  of  tho  triangle,  AOQ,  —  doul)le  the  area  of  the 
iiiangle,  AOB,  since  the  two  triangles  have  (lie  same  base, 
AO.  and  are  between  the  same  parallels.  AO  and  Qli. 
Uonce  the  moment  oi  AQ  about  0  —  the  moment  of  AH 


70 


VAKIONOJV'S   THEOIiEM  OF  MOMENTS. 


about  0  =  AB  •  p.  Therefore  the  sum  of  the  moments  of 
AP  and  AQ  about  0  =  the  sum  of  the  niononts  of  AV 
and  AB  about  0  =  {AV  +  AB)p,  =  {AB  -\-  BR)p, 
(since  AC  =  BR  from  tlie  equal  triangles  .IPC and  QBR) 
=  .1/^  '21  =  the  moment  of  the  resultant. 

If  the  origin  of  nioments  fall  bettueen  AP  and  AQ,  tht 
foi-ees  will  tend  to  jjroduce  rotation  in  oj)posite  directions  - 
and  hence  their  moments  will  have  contrary  signs  (Art. 
47).  In  this  case  the  moment  of  the  resultant  =  the  dif- 
ference of  the  moments  of  the  components,  as  the  student 
will  find  no  diiliculty  in  showing.  Hence  in  either  case 
tlie  moment  of  'l^e  resultant  is  equal  to  the  algebraic  sum 
of  the  moments  of  the  components. 

CoK.  1. — li"  tliere  are  any  number  of  component  forces, 
(V'c  may  compound  thorn  in  order,  taking  any  two  of  tliem 
!lrs't,  tiien  finding  the  resuUant  of  these  two  and  a  third, 
iV.id  so  on;  a  l  it  follows  that  the  sum  of  their  moments 
(with  their  projier  signs),  is  equal  to  the  moment  of  the 
resultunt. 

Cor.  2.— If  the  origii;  of  moments  be  on  the  line  of 
action  of  the  resultant,  p  =  0,  and  therefore  the  moment 
of  the  resultant  ~r.  0  ;  hence  the  sum  of  the  moments  of 
tiie  components  is  equal  to  zero.  In  this  case  the  moments 
of  the  forces  in  one  direction  bahmce  those  in  the  opposite 
direction ;  i.  c,  tlic  forces  that  tend  to  produce  rotation  in 
one  dire(!tion  ai'e  couiiteracted  by  the  forces  that  tend  to 
])roduce  rotation  in  the  opposite  direction,  and  there  is  no 
tendency  to  rotation. 


Cor.  3. — If  all  the  forces  are  in  equilibrium  the  resultant 
A'  ™  0,  and  Hierefore  tiic  moment  of  A'  =  0;  hence  the 
sum  (if  the  inonu'nts  of  the  components  is  ecpial  to  zero, 
and  llicre  is  no  t(;ndency  to  motion  eitii.-r  of  tmuHlatiou  or 
rotation. 


ra. 


VAJiIG.\0.\'S  TUEOREM  FOli   PARALLEL  FORCES.      71 


moments  of 
■onts  of  J  (,' 
B  +  BR)i), 
C'aud  QBIi) 

and  AQ,  tliL 
e  directions- 
(  signs  (Art. 
e  =  the  dif- 
the  student 
1  either  case 
Igelraic  sum 


)nent  forces, 
two  of  tlieni 
and  a  third, 
eir  moments 
ment  of  the 


I  the  line  of 
the  moment 

moments  of 
the  moments 
tlio  oj)posite 
J  rotation  in 
that  tend  to 

there  is  no 


the  resultant 
) ;  hence  the 
[ual  to  zero, 
'auulutiou  or 


Coil.  4. — Therefore  when  the  moment  of  the  resultant 
=  0,  wo  conclude  either  that  the  resultant  =  0  (Cor.  3), 
or  that  it  ^^assjs  through  the  point  taken  as  the  origin  of 
moments  (Cor.  2). 

58.  Varignon's  Theorem  of  Moments  for  Parallel 
Forces. — Th,o  aitnh  of  the  moments  of  tivo  jnirallei 
forces  about  any  point  is  equal  to  the  moment  of 
their  resultant  about  the  poitit. 


Let  P  and  Q  be  two  'parallel  forces 
acting  at  A  and  B,  and  R  their  result- 
ant acting  at  G,  aiid  let  0  he  tlie  point 
about  which  moments  are  to  be  taken. 
Then  (Art.  45)  we  have 


♦R 


♦<2 


Fig. 23 


P  X  AG  =  <2  X  BG, 
.-.    P(OG  -OA)  =  Q  (OB  -  OG), 
.-.       (P  +  0  OG  =  P  X  OA  +  ^  X  OB, 

7?  X  OG  =  P  X  OA  +  e  X  OB; 

that  is,  the  sum  of  the  moments  =  the  moment  of  the 
resultant. 

Cor. — It  follows  that  the  algebraic  sum  of  the?  moments 
of  any  number  of  parallel  forces  in  one  plane,  with  respect 
to  a  point  in  their  plane,  is  equal  to  the  moment  of  their 
resultant  with  respect  to  tlie  point. 

59.  Centre  of  Pai-allel  Forces. —  To  find  the  mag- 
iiitiidc,  (1  irt'ction ,  ami  point  of  applical ion.  of  the 
resultant  of  niiij  nnniJter  of  parallel  forces  acting 
vn,  a  rigid  body  in  on,e  plane. 


72 


CENTRE  OF  PARALLEL   FORCES. 


d 


Let  P^,  i'g,  P3,  etc.,  denote  the 
forces,  i/j,  M^,  M^,  etc.,  their  points 
of  applitiition.  Take  any  point  in 
the  plane  of  the  forces  as  origin  and 
draw  the  rectangular  axes  OX,  OF. 
Let  {x^,  y,),  \^^,  y^),  etc.,  be  the  "  *  »  "  * 
|)oints  of  application,  My,   i/g,    etc.  ''" 

Join  i/jJ/g;  and  take  the  point  i¥on  M^M^,  so  that 

M^M Pg 


M, 


il/,3/g  ~  F^+l 


rj    J 


0) 


then  the  resultant  of  /\  and  Pg  is  P,  +  Pg,  and  it  acts 
through  M  parallel  to  P,  (Art.  45). 

Draw  M^a,  Mb,  M^c  parallel,  and  M^e  perpendicular  to 
the  axis  of  y.    Then  we  have 

Mi^L  -  ^^'^  -  ¥]l-i1i  . 

i/,7l/g   ~  il/gC  ~"    2/g  -2/1  ' 


.'.     Mb-y^=-jr~^-rr{y2-yi)'> 

^  1   "r  ^  2 


Pi^i  +  Pg^a 


which  gives  the  ordinate  of  the  point  of  ai^plication  of  the 
resultant  of  1\  and  f  g. 

Now  since  the  resultant  of  P,  and  Pg,  which  is 
Pi  4-  I\,  acts  at  M,  the  resultant  of  J\  +  Pg  at  M,  and 
P3  at  iVg,  is  Pi  +  Pg  +  Ps  at  /;,  and  substituting  in  (x') 
Pj  +  Pg,  P3,  i/i,  and  y^  for  Pi,  Pg,  7/,,  and  i/g  resixn- 
tively,  we  have 

,  _  (''1  +  i\)  Mb  +  P^tu  „   i\y^±£^^li  +  ^\y-^ .  r!i 


p/m 

^ 

''m,    / 

*^ 

r 

i/      M. 

d 

e 

6  "A 
Fig.24 

J,  SO  that 


0) 


^2,  and  it  acts 
rpendicular  to 


2/i); 


2/8 


(3) 


lication  of  the 

Pg,    which    is 

Pg  at  M,  and 

titiiting  in  (x*) 

and  y^  resi)W' 


;(:!) 


CENTRE  OF  PARALLEL  FORCES. 


73 


and  this  process  may  be  extended  to  any  nnmber  of  parallel 
forces.  Let  R  denote  the  resultant  force  and  y  the  ordi- 
nate of  the  point  of  application  ;  then  we  have 

R  =  I\  +  Pg  +  1\  +  etc.  =  SP. 


y  = 


PilU  +  ^2l2jL:?3/^3__+ etc.  _  SPy 


I\  +  Pg  +  P3  +  etc. 


i:P 


Similarly,  if  i«  be  the  abscissa  of  the  point  of  application  of 
the  resultant,  we  have 


X  = 


-LPx 


The  values  of  x,  y  are  independent  of  the  angles  which 
the  directions  of  the  forces  make  with  the  axes.  Hence 
if  these  directions  be  turned  about  the  points  o  iiS''"'-'ation 
of  the  forces,  their  parallelism  being  preserved,  i  int  of 
application  of  the  resultant  will  not  move.  For  this  reason 
the  point  {x,  y)  is  called  the  centre  of  parallel  forces.  We 
shall  hereafter  have  many  ai)plications  in  which  its  position 
is  of  great  importance. 

ScH.  1. —  The  moment  of  a  force  with  respect  to  a  plane. 
is  the  product  of  tlie  force  into  the  perpcmlicnilar  distance 
of  its  point  of  application  from  the  plane.  Thus,  /\y,  is 
the  moment  of  the  force  P,,  in  reference  to  the  plane 
through  OX  perpendicular  to  OF.  This  must  be  carefully 
distinguished  from  the  moment  of  a  force  with  respect  to 
a  point.  Hence  the  equations  for  determining  the  position 
of  the  centre  of  parallel  forces  show  that  the  sum  of  t/w 
moments  of  the  parallel  forcer  with  respect  to  any  i)lane,  is 
equal  to  the  moment  of  their  resultant. 

Son.  2. — The  moment  of  a  force  with  respect  to  any  line 
is  the  product  of  the  component  of  the  force  perpendicular 
4 


71 


CONDITIONS   OF  EqVlLIUHIUH. 


Fig.25 


tf, 


CO  tbo  liijo  into  the  sliortost  distance  between  tlie  line  and 
the  line  of  action  of  the  force. 

60.  Conditions  of  Equilibrium  of  a  Rigid  Body 
acted   on   by   Parallel   Forces   in  one  Plane.— Lot 

1\,  I\,  7*3,  etc.,  denote  the  forces.  Take 
any  jxiint  in  tiie  plane  of  the  forces  as 
origin,  and  draw  rectangular  axes,  OX. 
OY,  the  latter  parallel  to  the  forces.  Let 
A  be  the  point  where  OX  meets  tlie  direc- 
tion of  /",,  and  let  OA  ~  x^. 

Apply  at  0  two  opposing  forces,  each 
equal  and  parallel  (o  P, ;  this  will  not  disturb  the  equili- 
brium. Then  I\  at  J  is  replaced  by  I\  at  0  along  OY, 
and  a  couple  whos(  moment  is  P^  ■  OA,  i.  c,  P  x  .  The 
renuiining  forces,  /%,  /'j,  etc.,  may  oo  treated  in  like  man- 
ner. We  thus  obtain  a  set  of  forces,  P^,  P^,  p  ^  etc. 
acting  at  0  along  OY,  and  a  set  of  couples,  /^.r,,  P^x  , 
P^x^,  etc.,  in  the  plane  of  the  forces  tending  to  turn  the 
body  from  the  axis  of  x  to  the  axis  of  y.  These  forces  are 
equivalent  to  a  single  resultant  force  7'j  ^  p^j^-p  ^  etc, 
and  the  couples  are  equivalent  to  a  single  resultant  couple, 
Pi^\  +  ^2-^2  4-  P^x^  +  etc.  (Art.  55). 

Hence  denoting  the  resultant  force  by  R,  and  the  moment 
of  the  resultant  couple  by  O,  we  have 

i?  =  Pj  -I-  Ps,  +  /"a  +  etc.  =  SP; 
0  =  Pi^i  +  P^x^  +  P^x^  -j-  etc.  =  SPa;; 

that  is,  a  system  of  jiarallel  forces  can  bo  reduced  to  a 
single  force  and  a  coui)lo,  which  (Art.  54,  Cor.)  cannot 
produce  e(|uilibrium.  ironci'.  for  o(iuilibriuni,  the  force 
und  the  couj)lo  nuist  vanish  ;  or 


SP  =  0,    and    ^Px  -.  0. 


the  line  and 


ligid  Body 
Plane.— Lot, 


Fig.25 

1 

)  the  equili- 
)  along  OV, 
P,x^.  The 
in  like  man- 
2»  -^  3>  ^"-^v 

to  turn  tlio 
se  forces  are 
■f  P3  +  etc., 
;ant  couple, 

the  moment 


(lunod   to  a 
'or.)  cannot 

I,   the  force 


COND'TIONH   OF  EQUILIBBIUM. 


<0 


1Y( 


>R 


-♦X, 


Fig.26 


Hence  the  conditions  of  eiiuilibrium  of  a  system  of  par- 
illel  forces  acting  on  a  rigid  body  in  one  plane  are  : 

17ie  sum  of  the /orc<-'s  mud  =  0. 

The  sum  of  the  moments  of  the  forces  about  every  jwint  in 
their  plane  must  =  0. 

61.  Conditions  of  Equilibrium  of  a  Rigid  Body 
acted  on  by  Forces  in  any  direction  in  one  Plane.  — 

Lot  1\,  Pg,  P3,  etc.,  be  tiie  forces  acting  at  the  points 

(•^i>  l/i),  (^8.  ^2).  (^3'  yw>  etc.,  in  the 
plane  xy.  Resolve  the  force  P^  into  two 
components,  JTi,  Fj,  parallel  to  OX 

and  OY  respectively.     Let  the  direc-     « 

tion  of  Fj  meet  OX  at  M,  and  the 
direction  of  X^  moot  OF  at  X.  Apply 
at  0  two  op[)osing  forces  each  equal  and  parallel  to  Xi, 
and  also  two  oi)posing  forces  each  ocpial  and  parallel  to  Fj . 
Hence  Fi  at  A^,  or  M,  is  etpiivalent  to  Fj  at  0,  and  a 
couple  whose  moment  is  F^  •  OM;  and  X^  at  Jj,  or  X,  is 
equivalent  to  X^  at  0,  and  a  couple  whose  moment  is 
X,  •  ON. 

Hence  F^  is  replaced  by  F,  at  0,  and  the  couple  Y^Xi  ; 
and  X^  is  replaced  by  X^  at  0,  and  the  couple  X^y^  (Art. 
47).  Therefore  the  force  Pj  may  be  rejilacod  by  the  com- 
ponents X^,   Fi    acting    at    0,  and    the    couple    whose 

moment  is 

Fi^i  -  X,yi, 

and  which  equals  the  moment  of  Pj  about  0  (Art.  57). 

liy  a  similar  resolution  of  all  the  forces  we  shall  have 
them  replaced  by  tlio  forces  (X^,  V.),  {A\,  F3),  etc., 
acting  at  0  along  the  axes,  and  tlie  (oui)les 


Fj.'g  —  A  g/Zj,      ) 


3' 3 


X^y.,,  etc. 


Adding  togetiier  the  couples  or  moments  of  Pj,  Pg,  etc., 


76  EQVILlBIilUM    IWVKR    THREE    FORCES. 

and  denoting  by  G  the  moment  of  the  resultant  couple,  wc 
get  the  total  moment 

G=:I.{Yx-Xy). 

If  the  sum  of  the  components  of  the  forces  along  OX  is 
denoted  by  SX,  and  the  sum  of  the  components  along  OY 
by  i  1",  the  resultant  of  the  forces  acting  ut  0  is  given  by 
the  equation 

Ri  =  {^Ay  +  (sr)2. 

If  a  be  the  angle  which  li  makes  with  the  axis  of  X,  we 
.lavo 

Ji  cos  a  =  iA",     li  am  a  =  :^Y; 

.'.    tan  a  =  — ^• 

Therefore,  any  system  of  f  >rces  acting  in  any  direction 
in  one  i)lane  on  a  rigid  body  may  be  reduced  to  a  single 
force,  E,  and  a  single  cou{)le  whose  moment  is  G,  which 
(Art,  51,  Cor.)  cannot  produce  equilibrium.  Hence  for 
equilibrium  wc  must  have  li  =  0,  and  6*  =  0,  which 
requires  that 

XX  =  0,    2:Y=  3, 

^{Yx-  Xy)  -  0. 

Hence  the  conditions  of  equilibrium  for  a  system  of 
forces  acting  in  any  direction  in  one  plane  on  a  rigid  body 
arc  : 

T/ie  sum  nf  the  componrnfs  of  the  forces  parallel  to  each  of 
tiro  rvctaiiijular  axes  must  =  0. 

The  sum  of  the  momeids  of  the  forces  round  every  point  in 
their  plane  must  —  0, 


at  coujjle,  we 


KXAHPhKS. 


77 


along  OX  is 
ts  along  OY 
•  is  given  by 


ixis  of  X,  wc 


my  direction 

to  a  single 

is  G.  whicii 

Hence  for 

=  0,   which 


1  system  of 
rigid  body 

el  to  each  of 
".ry  pomt  it, 


Cor. — Conversely,  if  i\\o  forces  arc  in  er|uilil)riuni  the 
sum  of  the  components  of  the  forces  parallel  to  any  direc- 
tion will  =  0,  and  also  the  sum  of  the  moments  of  the 
forces  about  any  point  will  =  0. 

62.  Condition  of  Equilibrimn  of  a  Body  under  the 
Action  of  Three  Forces  in  one  Plane. — //'  tlirer 
forces  DKiintalii  a  hndij  in  eqitilihriuin,  their 
direotions  Diitst  Dieet  in  (i  point,  or  he  parallel. 

Suppose  the  directions  of  two  of  the  forces,  /*  and  Q,  to 
meet  at  a  point,  and  take  moments  round  this  point ;  then 
the  moment  of  each  of  these  two  forces  =  0;  therefore  the 
moment  of  the  third  force  li  =  0  (Art.  01,  Cor.),  which 
requires  either  that  Ji  =  0,  or  that  it  pass  through  the 
point  of  intersecti(m  of  P  and  Q.  If  R  is  not  =  0,  it  must 
pass  through  this  jtoint.  Hence  if  any  two  of  the  forces 
meet,  the  third  must  pass  through  their  point  of  intersec- 
tion, and  keep  it  at  rest,  and  each  force  must  ho  equal  and 
opposite  to  the  resultant  of  the  other  two.  ^f  the  angles 
between  them  in  pairs  be  p,  q,  r,  the  forces  must  satisfy  the 
conditions 

P  :  Q  :  E  =  sin  p  :  sin  q  :  sin  r  (Art.  32). 

If  two  of  the  forces  are  parallel,  the  third  must  be 
parallel  to  them,  and  equal  and  directly  opposed  to  their 
resultant. 

EXAMPLES. 

1.  Suppose  six  parallel  forces  proportional  to  the  numbers 
1,  2,  3,  4,  5,  G  to  act  at  points  (—2,  —1),  (  —  1,  0),  (0,  ]), 
(1,  2),  (2,  3),  (3,  4)  ;  find  the  resultant,  R,  and  the  centre 
of  parallel  forces. 

By  Art.  59  we  have 

R  =  I.P  =  \  +2  +  ...G  =  21; 


V8 


EXAMPLES. 

SPx  =  -2-2  +  4  +  10  +  18  =  28; 
^Py  =  _  1  +  ;5  +  8  +  15  +  24  =  49. 


^  ~    }uP    -  21'    y  ~  ~1.P 


49 
21* 


2.  At  the  three  vertices  of  a  triangle  parallel  forces  are 
applied  which  are  jjroportioi.al  respectively  to  the  opposite 
sides  of  the  triangle;  find  the  centre  of  these  forces. 

Let(a-,,  ^i),  (xg,  ijg),  {Z3,  y^)  be  the  vertices,  and  let  a, 
b,  c  be  the  sides  opposite  to  them ;  then 

fl  +  *  +  c       '   ^  a  +  b  +  c 

3.  If  two  parallel  forces,  P  and  Q,  act  in  the  same  direc- 
tion at  A  and  B,  (Fig.  14),  and  make  an  angle,  0,  with 
A  B,  find  the  moment  of  each  about  the  point  of  applica- 
tion of  their  resultant. 

The  moment  of  P  with  respect  to  O  is 

P-^G'sinOCArt.  46). 
But  from  (1)  of  Art.  45,  we  have 

P+  Q  _AB 


Q 

.'.    AG  = 

which  j     P'  AG  sin  d  gives 
PQ 


AG' 
Q 


P  +  Q 


AB, 


AB  sin  e, 


P^Q 
for  the  moment  of  P  which  also  equals  the  moment  of  Q 


9. 

to 
n 

1  forces  are 

;he  opposite 

rces. 

,  and  let  a, 


same  direc- 

gle,  0,  with 

of  applica- 


ent  of  Q 


•m'' 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


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1.0 


I.I 


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12.5 
12.2 

M 

1.8 


1.25 

1.4       1.6 

4 6"     

► 

•m 


PhotogKiphic 

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^m 


EXAMPLKS. 


79 


4.  Two  parallel  forces,  acting  in  the  same  direction, 
liave  tlioir  iniigiutndcH  5  and  \'-\,  and  their  points  of  appliea- 
lion,  A  and  //,  0  leet  apart.  Find  the  magnitude  of  their 
resultant,  and  tlie  point  of  application,  (I. 

Ans.   R  =  18,  A(J  =  4^,  BG  =  If. 

5.  On  a  straight  rod,  AF,  there  are  suspended  5  weights 
jf  5,  15,  7,  0,  and  9  pounds  respectively  at  the  points  A,  B, 
D,  E,  F;  AH  =  3  feet,  BD  —  G  feet,  DE  -  5  feet, 
EF  =  4  feet.  Find  the  magnitude  of  the  resultant,  and 
the  distance  of  its  point  of  application,  G,  from  A. 

Ans.  R  =  42  pounds.  AG  =^  ^^  feet. 

G.  A  heavy  uniform  beam,  Ali,  rests 
in  a  vertical  plane,  with  one  end,  A,  on  a 
smooth  horizontal  plane  aiul  the  other 
end,  li,  against  a  smooth  vertical  wall ; 
the  end,  A,  is  prevented  from  sliding  by 
a  horizontal  string  of  given  length  fas- 
tened to  the  end  of  the  beam  and  to  the  wall ;  determine 
the  tension  of  the  string  and  the  pressures  against  the 
horizontal  plane  and  the  wall. 

Let  "Za  =  the  length  of  the  beam,  and  let  W  be  its  weight, 
which  as  the  beam  is  uniform,  we  may  suppose  to  act  at  its 
middle  point,  G.  Let  R  be  the  vertical  pressure  of  the 
horizontal  plane  against  the  beam  ;  and  R'  the  horizontal 
pressure  of  the  vertical  wall,  and  T  the  tension  of  the  hor- 
izontal string,  AC  ;  let  liAC  =:  «,  a  known  angle,  since 
the  lengths  of  the  beam  and  the  string  are  given.  Then 
(Art.  Gl),  we  have 

for  horizontal  forces,  T  =  R' ; 


Fia.27 


for  vertical  forces,      W  =  R ; 

for  moments  about  A  (Art.  47),  2R' a  sin  « 

W 
.-.     A"  =  7'=-^  cot  a. 
4 


:  Wn  cos  «  ; 


mm 


80 


EXAMPLES. 


7.  A  heavy  beam,  A  B  =  r/  +  />,  rests 
oil  two  givtMi  sniootli  iiliinos  which  are 
iiK'lincd  iit  angles,  «  and  fi,  to  the 
horizon;  n'i|nirt'(l  the  angle  0  wiiich 
the  beam  makes  with  the  horizontal 
phme,  and  the  pressures  on  the 
phmes. 

Let  a  and  i  be  the  segments,  AG  and  BU,  of  t!ic  beam, 
made  by  its  centre  of  gravity,  G ;  let  I'  and  R'  be  the 
pressures  on  the  planes,  AC  and  BC,  the  lines  of  action  of 
which  are  perpendicular  to  tiie  planes  since  they  are  smooth, 
and  let  IT  be  the  weight  of  the  beam.     Then  we  have 

for  horizontal  forces,  E  sin  «  =  li'  sin  |8;  (1) 

for  vertical  forces.   A'  cos  «  +  Ji'  cos  0  =  W;       (2) 

for  moments  about  G,  Ra  cos  {a  +  &)  =  H'h  cos  (0—6).  (3) 

Dividing  (.'{)  by  (1),  we  have 

a  cot  a  —  a  tan  0  =  b  cot  0  +  b  tan  8  ; 

a  cot  «  —  b  cot  0 


therefore, 


tan  0  = 


a  +  b 


and  from  (1)  and  (2)  wo  have 
W  sin  3 


R  = 


sin  («  -f  0) 


;    and  R' 


W  Bin  a 

sinXfT+^y 


Otherwise  thus:  since  the  beam  is  in  equilibrium  under 
the  action  of  only  three  forces,  they  must  meet  in  a  point  (), 
(Art.  (i2),  and  therefore  wo  obtain  immediately  from  the 
geometry  of  the  figure, 


R  _       sini3 
W~  8in(«  +  j3)' 


/.' 


W8m0 
8in'V«  +  0)' 


T,  of  t!ic  beam, 
and  R'  be  tlie 
los  of  action  of 
liey  are  smooth, 
I  Ave  have 

sini8;  (!) 

(i=W;       (2) 

cos  (/3-d).  (3) 

an  6 ; 


sin  a 

ilibriiim  under 
:'et  in  a  iioiiit  (), 
ately  from  tiie 

sin  (i 


EXAMPLES. 


81 


and 


R' 


sm  « 


•.         A"=-T 


TFsin  « 


ir       sin  [a  -\-  liy         '  '     "        sin  («  +  |3) 

Also  since  the  angles,  (JO A  ami  (JOI?,  are  equal  to  «  and  |3, 
respectively,  and  BGO  —  ^  —  0,  we  liave 


therefore, 


(rt  +  i)  cot  JJGO  z=  a  cot  GOA  -  b  cot  GOB; 
a  cot  a  —  h  cot  /3 


tan  0  = 


«  +  * 


Hence,  if  -j  =  -' — ^-,  the  beam  will  rest  in  a  liorizontal 
0       tan  ji 

position. 

8.  A  heavy  -nniform  beam,  AB,  rests  with 
one  end  A,  against  a  smooth  vertical  wall, 
and  the  other  end,  B,  is  fastened  by  a  string, 
BC,  of  given  length  to  a  point,  (',  in  the 
wall ;  the  beam  and  the  string  are  in  a  vertical 
l)iane ;  it  is  required  to  determine  tlie  pressure 
against  tho  wall,  the  tension  of  the  string,  and 
the  position  of  the  beam  and  the  string. 


Let 


AG  =  GB  =.  a,    AC  =  .r,     BC  =  b, 


weight  of  beam  =  W,  tension  of  string  =  T,  pressure  of 
wall  =  R, 

BAE  =  0,    BCA  =  ^. 
Then  we  have 

for  liorizontal  forces,  7?=  Tm\^;  (1) 

for  vertical  forces,       ]V  =  T  cos  0 ;  (3) 

for  moments  about  A,  Wa  sin  0  =  T-  AD  =  T.r.  sin  0;  (.1) 

.'.    a  sin  0  ■—  X  tan  0;  (4) 


^ 


82 


EXAMPLES. 


and  by  the  geometry  of  tlie  figure 

b    _  sill  0  ^ 
Ha       siu^' 

X  _  sin  (9—4) 
2a  sin  </> 

Solving  (4),  (5),  and  (0),  we  get 

"  =  L-ir-J ' 


COS 


^       2  r^a  -  ia^y 


sin  6  — 


2a 


3 


T' 


(5) 


(6) 


from    which    7?    and    T  become  known.     (Price's  Anal. 
Mech's.,  Vol.  I,  p.  flO). 

To  determine  all  the  unknown  quantities  many  problems  in  Statics 
require  equations  to  be  formed  by  geometric  relations  as  well  as  static 
relations.  Thus  (1),  (3),  (3)  are  static  equations,  and  (5)  is  a  geometric 
equation. 

9,  A  uniform  heavy  beam,  AB  =  2a, 
rests  with  one  end,  A,  against  the  inter- 
nal surface  of  a  smooth  iiomi8i)herical 
b;)wl,  radius  =  r,  while  it  is  supported 
at  some  point  in  its  length  l)y  the  edge 
of  the  bowl ;  find  tho  position  of  equili- 
brium. 

The  beam  is  kept  in  equilibrium  by  three  forces,  viz.,  tho 
reaction,  R,  at  A  perpendicular  to  the  surface  of  contact, 
(Art.  42)  and  therefore  ]icrpendicular  to  the  bowl,  tlio 
r>,>action,  R,  at  C  which,  for  the  same  reason,  is  per])eti- 
dicular  to  the  beam,  and  tlie  weight  U' acting  at  U, 


Fig. 30 


(6) 
(6) 


(Price's  Anal. 

|)roblems  in  Statics 
J 8  as  well  as  static 
1  ^5)  is  a  geometric 


Fig. 30 


forces,  viz.,  the 
rface  of  con  tart, 
the  howl,  the 
ison,  is  {)t'r])eti- 
ig  lit  U, 


EXAMPLES. 


8:i 


Let  0  =  the  inclination  of  tlie  beam  to  the  horizon 
=  <ACI).  The  solution  will  be  most  readily  effected  liy 
resolving  the  forces  along  the  beam  and  taking  moments 
iil)ont  C,  by  which  we  shall  obtain  equations  free  from  tise 
unknown  reaction,  7?'.     Then  we  have 


for  forces  along  AB,  Jt  cos  0  =  W  sin  0, 

for  moments  about  C, 

R  •  2r  cos  d  sin  d  =  W  {2r  coe  6  —  «)  cos  d. 

From  (1)  we  have 

B  =  W  tan  e, 

which  in  (2)  gives,  after  reducing, 

2/-  sin'^  0  —  2r  cos''  d  +  acosO  =  0, 

4r  cos^  0  —  a  cos  0  —  'ir  =:  0, 


(1) 
(2) 


or. 


(3) 


cos  6  = 


8/- 


Otherwise  thus:  since  the  beam  is  in  or; iiilibrium  under 
tiie  action  of  only  three  forces,  tliey  must  cieet  in  a  point 
0  (Art.  GH).  Draw  the  tiireo  forces  AC,  (W,  GO,  which 
keep  the  beam  in  equilibrium.  Let  the  line,  GG,  meet  the 
semicircle,  DAC,  in  the  point.  Q.  Thou  AQ  is  a  horizontal 
line.     Also 

<QAG  =  <DCA  =  6, 


therefore 
I I en CO 
and  also 


<OAQ  =  20. 
AQ  =  AO  cos  26, 
AQ  =  AG  cos  0; 


m 


84 


EXAMPLES. 


therefore 


or 


2r  cos  2d  =  a  cos  8, 
4r  cos^  0  —  «  cos  0  —  2r  =  0, 


wliich  is  the  same  as  (3)  obtained  by  tlio  other  method, 
'rhe  student  may  prove    that    the    reaction,    W,  at  C 
ffl 


=  W 


2r 


Fig. 31 


rw 


10.  Find  the  position  of  equilibrium  of 
a  uniform  heavy  beam,  one  end  of  which 
rests  against  a  smooth  vertical  plane,  and 
the  other  against  the  internal  surface  of  a 
smooth  spherical  bowl. 

The  beam  is  in  equilibrium  under  the 
action  of  throe   forces,    the    weight,   W, 
acting  at  G,  the  reaction,  J?,  at  A,  perpen- 
dicular to  the  surface  and  hence  passing  through  the  centre, 
C,  and  the  reaction,  E',  of  the  vertical  jdane  perpendicular 
to  itself  and  benec  horizontal. 

Let  the  length  of  tlie  I)oam,  AH,  =  2a,  r  =  the  radius 
of  the  sphere,  d  =  CD,  tlie  distance  of  the  centre  of  the 
sphere  from  the  vertical  wall,  IT  =  the  weight  of  the  beam  ; 
and  let  0  =  the  required  inclination  of  the  l»eam  to  the 
horizon,  and  0  =  the  inclinaliou  of  the  radius  A('  to  the 
horizon.     Then  we  have 

for  vertical  forces,  R  sin  (p  z=  W ;  (1) 

for  moments  about  B,  /?•  2a  sin  {(p—O)  =  W-acosO;  (2) 

Dividing  (3)  by  (1)  wo  liave 

2  sin  (0  —  6) 


=r  cos  9, 


or 


sin  (j) 
tan  <p  =  2  tan  9. 


(3) 


Dr  method, 
tion,    W,  at  C 


Fig. 31 


lugh  t  lie  centre, 
3  pcrpL'ntliciiliir 

/•  :=  the  radius 
le  centre  of  the 
it  of  the  heam  ; 
e  heam  to  the 
diuH  A("  to  the 


V;  (1) 

IF-fflCosO;  (2) 


(3) 


CENTRE    OF   PARALLEL    FORCES. 


85 


Then  wc  have,  from  the  geometry  of  the  figure,  the 
horizontal  distance  from  A  to  tiie  wall  =  the  horizontal 
projection  af  AC  +  CD,  that  is. 


2a  cos  0  =  r  cos  0  +  d. 


(4) 


From  (3)  and  (4)  a  value  of  d  can  be  obtained,  and  hence 
tile  position  of  equilibrium. 

Otherwise  thus:  since  the  beam  is  in  e(inilibrium  under 
the  action  of  only  three  forces  they  must  meet  in  a  point,  0. 
Geometry  then  gives  us 

2  cot  0GB  =  cot  AOG  -  cot  GOB  =  cot  AOG, 


or 


2  tan  9  —  tan  0, 


which  is  the  same  as  (.3). 

63.  Centre  of  Parallel  Forces  in  Different  Planes. 

— 2'()  find  the  magnittidc,  direction,  and  point  of 
a/)f)li cation  of  the  resultant  of  any  number  of 
parallel  forces  acting  on  a  rigid  body. 

The  theorem  of  Art.  59  is  evidently  true  also  in  the  case 
in  which  neither  the  parallel  forces  nor  their  fixed  points  of 
application  lie  in  tiie  same  plane,  hence,  calling  i  the  third 
co-ordinate  of  the  point  of  application  of  the  resultant,  we 
have  for  the  distance  of  the  centre  of  parallel  forces  from 
the  planes  yz,  zx,  and  xy, 


X  =: 


_  iP^      -  _  I.Py      .  _  SP« 


^P 


i/ 


J  > 


e  = 


^r 


Hence  (Art.  59,  Sch.)  the  equations  for  determining  the 
])osition  of  the  centre  of  parallel  forces  show  that  the  sum 
I'f  llie  nionien/s  of  the  parallel  forces  with  respect  to  any 
plane  is  equal  to  the  moment  of  their  resultant. 


86     KQUILIBHIVM   OF  PARALLEL   FORCES  L\  SPACES. 

64.  Conditions  of  Equilibrium  of  a  System  of 
Parallel   Forces   Acting   upon  a    Rigid    Body   in 

Sp-^ce.— Let  1\,  l\,  l\,  etc.,  denote  the  forces,  and  let 

them  be  referred  to  three  rectangular  axes, 

0:C  or,   OZ;    the   last  parallel   to  the 

forces  ;   let  (a;,,  y„  z^),  {x^,  y„,  z^),  etc., 

be  the  points  of  ap2)lication  of  the  forces, 

1\,  7*2,  etc.      Let  the   direction   of  I\ 

meet  the  plane,  xy,  at  J/, . 

Draw  M^N^  perpendicular  to  the  axis 
of  X  meeting  it  at  X^.  Apply  at  0,  and  also  at  N^,  two 
opposing  forces  each  equal  and  parallel  to  I\.  Then  the 
force  Pj  at  Jf,  is  replaced  by 

(1)  P,  at  0  along  OZ; 

(2)  a  couple  formed  of  I\  at  J/,  and  P,  at  N^  ; 
(:i)  a  couple  formed  of  P,  at  N^  and  1\  at  0. 

The  moment  of  the  first  couple  is  P^y^,  and  this  couple 
may  be  transferred  to  the  plane  yz,  which  is  parallel  to  its 
original  plane,  without  altering  its  effect  (Art.  52).  The 
moment  of  the  second  couple  is  P^x^,  and  the  couple  is  in 
the  plane  xz. 

Replacing  each  force  in  this  manner,  the  whole  system 
will  be  equivalent  to  a  force 

Pi  +  Pg  +  P3  +  etc.,  or  SP  at  0  along  OZ, 
together  with  the  couple 

^12/1  +^2^2  + -Psys+etc,  or  ILPy,  in  the  plane  yz, 

and  the  couple 

Pj^i  +  Po-Tj  -I- Pj.r J  +etc.,  or  SPa;  in  the  plane  u-z. 

The  first  couple  tends  to  turn  the  body  from  the  axis  of// 
'y  that  of  z  round  the  axis  of  x,  and  the  second  couple 


T 


■.V  SPACES. 

I  System  of 
i;id    Body    in 

tbrcos,  and  let 


ilso  at  iV, ,  two 
f\.    Then  the 


it  jV,  ; 
it  0. 

rid  this  couple 
parallel  to  its 
Art.  5a).  The 
ic  coujile  is  in 

whole  system 


igOZ, 


le  plane  yz, 


c  plane  aZ. 

u  tliL'  axis  of// 
second  couple 


EQUILIIiRIUM   OF  PARALLEL   FORCES  IN  SPACE.      87 

tends  to  turn  the  body  from  the  axis  of  x  to  that  of  z 
round  the  axis  of  y.  It  is  customary  to  consider  those 
t!)iipk's  as  i)ositive  which  tend  to  turn  the  body  in  the 
direction  indicated  by  the  natural  order  of  the  letters,  i.  e., 
poaitice  from  x  to  y,  round  the  2-axis ;  from  ^  to  z  round 
the  X-axis ;  and  from  z  to  x  round  the  /y-axis  ;  and 
vef/atire  in  the  contrary  direction. 

Hence  the  moment  of  the  first  couple  is  +2P//,  and 
therefore  OX  is  its  axis  (Art.  50) ;  and  the  moment  of 
the  second  couple  is  —'LPx,  and  therefore  OY'  is  its  axis. 
The  resultant  of  these  two  couples  is  a  single  couple  whose 
axis  is  found  (Art.  5G)  by  drawing  OL  (in  the  positive 
direction  of  the  axis  of  x)  —  '^i'y,  and  OM  (in  the  nega- 
tive direction  of  the  axis  of  y)  =  "^Px,  and  completing  the 
parallelogram  OLGM.  If  OG,  the  diagonal,  is  denoted  by 
G,  we  have 

G  =  y/WxfT'i^'y?, 


and 


R  =  2P; 


R  being  the  resultant  force. 

Now  since  this  single  force,  R,  and  this  single  couple,  G, 
cannot  produce  eqnilibrium  (Art.  54,  Cor.),  we  mi  st  have 
R  =  0,  and  G  =:  0,  i'.nd  G  cannot  be  =  0  unless  2/^a;  =  0 
and  ^Py  =  0 ;  the  conditions  therefore  of  equilibrium  axe 

i?  =  0, 

2P«  =  0,     ^Py  =  0. 

Hence,  the  conditions  of  equilibrium  of  parallel  forces  in 
space  are : 

The  sum  of  the  forces  must  =  0. 

The  sum  of  the  moiiiettts  of  the  forces  tvith  respect  to 
iccry  plahe  parallel  to  them  must  =.  0. 


88 


EQurLiniirvM  of  forces. 


Fig.33 


65.  Conditions  of  Equilibrium  of  a  System  of 
Forces  acting  in  any  Direction  on  a  Rigid  Body  in 
Space. — Lc'l  1\,  /'„,  l\,  'itc,  donote  <'ie  fci-ecs,  ami  let, 
them  bo  rofcrred  to  tlirco  rectangulur  axes,  OX,  01'.  OZ; 
lot  (.r,,  Vj,  z^),  {x^,  11^,  Zg),  etc.,  be  the  pointis  of  applica- 
tion of  P,,  P.^,  etc. 

Let  .Ij  be  tlie  point  of  application  of 
1\;  resolve  /'j  into  coniponent.s  X-^, 
I'j.  iTj,  jiarallel  to  the  co-ordinate  axes. 
Lot  the  direction  of  Z^  meet  the  \Ai\\\c 
xy  at  J/j,  and  draw  M^N-^  perpendicu- 
lar to  OX.  Apply  at  N^  and  also  at  0 
tyro  opposing  forces  each  equal  and  i)ar- 
allel  to  Zj.  Hence  Z^  at  A^  or  .1/",  is  ef[uivalent  to  Z^  at 
0,  and  two  coujdes  of  which  the  former  has  its  moment  = 
Z^  X  -Vj3/j  —  Z^iii,  and  may  be  supposed  to  act  in  the 
plane  yz,  and  the  latter  has  its  moment  =  Zj  x  ON^  = 
—  Z^x^  and  acts  in  the  jdane  zx. 

Hence  Z^  is  replaced  by  Z,i\i  0,  a  couple  Z-^y^  in  tlie 
plane  yz,  and  a  couple  —  Z^i\  (Art.  G-i)  in  the  plane  zx. 
Similarly  A'j  may  be  rejilaced  liy  A'j  at  0,  a  coujjle  X^z^ 
in  tlie  plane  zx,  and  a  couple  —  ^^i^i  i»  the  2)lane  xy. 
And  3^1  may  be  replaced  by  Y^  at  0,  a  cou]de  Y^x^  in  the 
])Iaue  xy,  and  a  couple  —  Y^z^  in  the  plane  yz.  Therefore 
the  force  /*j  nniy  be  rejjlaced  1)y  X^,  Y^,  Z^,  acting  at  0, 
and  three  couples,  of  which  the  moments  are,  (Art.  50), 

Zj^i  —  FjZj  in  the  piano  yz,  around  the  axis  of  .r, 
XjZj  —  Z^x^  in  the  i)lane  zx,  around  tlie  axis  oi y, 
3'j;/'i  —  X-^y^  in  tlie  jilane  xy,  around  the  axis  of  z. 

By  a  similar  resolution  of  all  the  forces  we  shall  havo 
them  replaced  by  the  forces 

-^x,   ir.    1^ 


acting  at  0  along  the  axes,  and  the  couj)les 


System  of 
igid  Body  in 

fci'fos,  iiiid  lot 
Ll,  UV.  OZ; 
its  of  upplica- 


f 

.z. 

X|. 

V 

A, 

N,    „ 

/ 

4< 

Fig.33 

iloiit  to  Z,  at 
ts  moment  = 
to  iict  in  tlu' 
JT,  X  ON,  = 

0  Z-^ii,  in  tlie 
tlic  piano  zx. 
coujjlo  ^V,^, 
tlie  plane  jy. 
e  iFi.r,  in  the 
z.  Therefore 
,  acting  at  0, 
(Art.  50), 

e  axis  of  .r, 
0  axis  of  y, 
c  axis  of  z. 

we  shall  havo 


EQUILIBRIUM  OF  FORCES.  89 

S  {Zy  —  J'z)  =  L,  suppose,  in  the  plane  yz, 
-  {Xz  —  Zx)  —  M,  ,sup})ose,  in  the  plane  zr, 
^^.{Yx  —  Xy)  =  N,  suppose,  in  the  plane  xy. 

Let  R  he  tl;o  resultant  of  th*^  .  rces  which  act  at  O;  a, 
I),  (',  the  angles  its  direction  makes  with  the  axes ;  then 
(Arti.  38), 

R^  =  {^xf  +  (ir)2  +  (:Lzy, 


cos  a 


^X 

It 


R  ' 


cos  b 


iF 


2Z 


R'     «««^  =  ^ 


Let  G  be  the  moment  of  the  couple  which  is  the  result- 
ant of  the  three  couples,  L,  M.  N ;  A,  fi,  v,  the  angles  its 
axis  makes  with  the  co-ordinate  axes  ;  then  (Art.  oG,  Sch.), 

6^2  =  D  +  J/2  +  N^, 

L  M  N 

cos  A  =  —;,    cos  /It  =  -^,     cos  V  =  —• 
b  i.T  Lt 

Therefore  any  system  of  forces  acting  in  any  direction  on 
a  rigid  body  in  space  may  always  be  reduced  to  a  single 
force,  R,  and  a  single  couple,  G,  and  cannot  therefore  i)rn- 
duce  equilibrium  (Art.  54,  Cor.).  Hence  for  equilibrium 
we  must  have  it'  =  0  and  ^r  =  0  ;  therefore 

(i:x)2  +  (i:r)2  +  (iz)2  =  o, 

and  7.2  ^  j/a  +  jy-a  _  o. 

These  lead  to  the  six  conditions, 

i;x=o,    i:F=o,    2:z=o, 

i;  {Zy  -  Yz)  =  0,     1  {Xz  -  Zx)  =:^  0, 
^{Yx  -Xy)  =  0. 


90 


EXAMPLES. 


EXAMPLES. 

1.  If  the  weights,  1,  2,  3,  4,  5  lbs.,  act  perpendicularly 
to  u  straight  line  at  the  respective  distances  of  1,2,  3,4, 
5  feet  from  one  extremity,  find  the  resultant,  and  the  dis- 
tance of  its  point  of  application  from  the  first  extremity. 

Ans.   R  —\b  lbs.,  .r  =  3f  feet. 

2.  Four  weights  of  4,  —7,  8,  —3  lbs.,  act  perpendicularly 
to  ti  straight  line  at  the  points  A,  B,  C,  D,  so  that  AB  = 
5  feet,  BC  =  4  feet,  CD  =  2  feet ;  find  the  resultant  and 
its  point  of  ai)plication,  G. 

Ans.  B  =  2  lbs.,  AG  =  2  feet. 

3.  Two  parallel  forces  of  23  and  42  lbs.,  act  at  the  points 
A  and  B,  14  inches  apart;  find  GB  to  three  places  of 
decimals.  Ann.  4  954  ins. 

4.  Two  weights  of  3  cwts.  2  qrs.  15  lbs.,  and  1  cwt.  3  qrs. 
25  lbs.  are  supported  at  the  points  A  and  B  of  a  straight 
line,  the  length  .VB  =  3  feet  7  inches ;  lii d'  AG  to  three 
})laces  of  decimals  of  feet.  Ans.   1.268  ft. 

5.  A  bar  of  iron  15  inches  long,  weighing  12  lbs.,  and  of 
uniform  thickness,  has  a  weight  of  10  lbs.  suspended  from 
one  extremity  ;  at  what  point  must  the  bar  be  supported 
that  it  may  just  balance. 

Tlie  weight  of  the  bar  acts  at  he>  centre. 

Ans.  i^  in.  from  the  weight. 

0.  A  bar  of  uniform  thickness  weighs  10  ll)s.,  and  is 
5  feet  long  ;  weights  of  9  lbs.  and  5  lbs.  are  suspeiuli'd  from 
its  extremities  ;  on  what  point  will  it  balance  ? 

Ans.  5  in.  from  the  centre  of  th<'  bar. 

7.  A  beam  30  feet  long  balances  itself  ow  a  point  ai  onc- 
thii'd  of  its  k'ngtli  from  Die  thicker  end  ;  but  when  a  'vi.jirlii, 
of  10  lbs.  is  suspended  from  the  smaller  end,  the  proj-  must 


EXAMPLES. 


91 


erpondicularly 
i  of  1,  ^,  3,  4, 
,  iuid  the  dis- 
extremity. 
■  ==  3f  feet. 

irpcndicnliirly 
50  tliiit  AB  = 
resultant  and 

J  =  2  feet. 

;  at  the  points 
liroe  places  of 
.  4  954  ins. 

1  1  cwt.  3  qrs. 
of  a  straight 
AG  to  three 

IS.   1.268  ft. 

12  Ihs.,  and  of 

spcnded  from 

be  supported 


the  weight. 

D  ll)s..  and   is 
spcndi'd  IVoin 

'  of  the  bar. 

point  -'li  one- 
wiicn  a  >v(.iglit 
,he  proj-  must 


be  moved  two  feet  towards  it,  in  order  to  maintain  the 
e(iuilibrium.     Find  the  weight  of  the  beam.  Ans.  90  lbs. 

8.  A  uniform  bar,  4  feet  long,  weighs  10  lbs.,  and  Aveights 
of  30  lbs.  and  40  lbs.  are  ai)pended  to  its  two  extremities  ; 
where  must  the  fulcrum*  be  placed  to  produce  equilibrium  ? 

Am.  3  in.  from  the  centre  of  the  bar. 

9.  A  bar  of  iron,  of  uniform  thickness,  10  ft.  long,  and 
weighing  1|  cwt.,  is  supported  at  its  extremities  in  a  hori- 
zontal position,  and  carries  a  weight  of  4  cwt.  suspended 
i'rom  a  point  distant  3  ft.  from  one  extremity.  Find  the 
pressures  on  the  points  of  support. 

Ans.  3.55  cwt.,  and  1.95  cwt. 

10.  A  bar,  each  foot  in  length  of  which  weighs  7  lbs., 
rests  upon  a  fulcrum  distant  3  feet  from  one  extremity ; 
Avhat  must  bo  its  length,  liuit  a  weight  of  71^  lbs.  sus- 
jtended  from  that  extremity  may  just  be  balanced  by 
20  lbs.  suspended  from  the  other?  Ans.  9  ft. 

11.  Five  equal  parallel  forces  act  at  5  angles  of  a  regular 
hexagon,  Avhose  diagonal  is  a  ;  find  the  point  of  application 
of  their  resultant. 

Ans.  On  the  diagonal  passing  through  the  sixth  angle,  at 
a  distance  from  it  of  |«. 

12.  A  body,  P,  suspended  from  one  end  of  a  lever  with- 
out weighty  is  balanced  by  a  weight  of  1  lb.  at  the  other 
end  of  the  lever  ;  and  when  the  fulcrum  is  removed 
through  half  the  length  of  the  lever  it  re(iui.res  10  lbs.  to 
balance  P  ;  lind  the  weight  of  P.      Ans.  5  lbs.  or  2  lbs. 

13.  A  carriage  wheel,  whose  weight  is  IF  and  radius  r, 
rests  u[)on  a  level  i-oad ;  show  that  the  force,  /•',  necessary 
to  draw  the  wheel  over  an  obstacle,  of  height  h,  is 

V2rF-A^ 


F=W 


h 


*  The  support  ou  wliich  it  rustB. 


m 


92 


EXAMPLES. 


14.  A  beam  of  uuifonn  tliicknotss,  5  feet  long,  woifiliiiig 
10  llxs..  is  supported  on  two  props  at  tlio  ends  of  the  beam  ; 
find  wliere  a  weight  ol'  'M)  lbs.  mii't  be  i)laeed,  so  that  the 
pressures  on  the  two  props  may  bn  15  lbs.  and  ^5  lbs. 

A)is.  10  ins.  from  the  centre. 

15.  Forces  of  3,  4,  5,  0  lbs.  lu-t  at  distances  of  ',\  ins., 
4  ins.,  5  ins.  (i  ins.,  from  the  end  of  a  rod  ;  at  what  distance 
from  the  same  end  docs  the  resultant  act';' 

Anf>.  4^  inches. 

10.  Four  vertical  forces  of  4,  0,  7.  0  lbs.  act  at  the  four 
corners  of  a  S(|uare  ;  lind  (lie  point  of  application  of  the 
resultant.     Ans.  ^'3  of  middle  line  from  one  of  the  sides. 

17.  A  flat  board  Vi  ins.  s(iiuire  is  suspended  in  a  hori- 
zontal position  by  strings  attached  to  its  four  corners,  A, 
B,  C,  D,  and  a  weight  eipud  to  the  wt  ight  of  the  lioard  is 
laid  upon  it  at  a  point  3  ins.  distant  from  the  side  AB  and 
4  ins.  from  AD  ;  lind  the  relative  tensions  in  the  four 
strings.  Aits.  As  f  :  |  :  J  :  ^^^. 

18.  A  rod,  AB,  moves  freely  about  the  end,  B,  as  on  a 

hinge.     Jts  weight,  IT,  acts  at  is  middle  point,  and  it  is 

kept  horizontal  l)y  a  string,  AC,  that  makes  an  angle  of  45° 

with  it.     Find  the  tension  in  the  string.  ,  11' 

^  Aim.  

19.  A  rod  10  inches  long  can  turn  freely  al)()ut  one  of 
its  ends  ;  a  weight  of  4  lbs.  is  slung  to  a  point  3  ins.  from 
this  end,  and  the  rod  is  held  l)y  a  string  attached  to  its  free 
end  and  inclined  to  it  at  an  angle  of  120°;  find  the 
tension  in  the  string  when  the  rod  is  horizontal. 

Alls.  ^  \/:\  lbs. 

•20.  Two  forces  of  .'Mbs,  and  -1  lbs.  act  at  the  extremities 
of  a  straight  lever  I'i  ins.  long,  and  inclined  to  it  at  angles 
of  VHf  and  135°  respectively;  lind  the  position  of  the 
fulcrum.  j„^.  (y  _  ;{  ^  ,j^  ^  9  (J  j„y^  jYoni  y„e  end. 


long,  \voij,'hiii|» 
s  of  tile  l)o;iin  ; 
•il,  8o  lliat  tliu 
il  ^5  lbs. 
[1  the  centre, 

IOCS  of  .'5  ins., 
t  what  (UsUuico 

;.  4^  inches. 

ct  lit  the  four 
lication  ot  the 
of  the  sides. 

lied  in  a  hori- 
iir  corners,  A, 
f  tile  hoard  is 
'  side  AB  and 
IS   in  the  four 

id,  B,  as  on  a 

oint,  and  it  is 
n  angle  of  45° 

Alls.  

'  ahoiil  one  of 

in(  ',i  ins.  from 

lied   (o  ils  free 

Uf  ;    find  the 

al. 

'.  J  V:\  ll)s. 

he  extreniit'cs 
o  it  al  aiigh's 
Ksition  of  the 
tm  one  end. 


EXAMPLES. 


93 


21.  Find  the  true  v/eight  of  a  body  which  is  found  to 
weigh  8  o/,s.  and  9  ozs.  wlien  placed  in  each  of  the  scale- 
pans  of  a  false  balance.  Ans.  0  V^  ozs. 

22.  A  beam  3  ft.  long,  the  weight  of  which  is  10  lbs., 
and  acts  at  its  middle  point,  rests  on  a  rail,  with  4  lbs.  iiang- 
iiig  from  one  end  and  13  lbs.  from  the  other ;  find  the  point 
at  wiiieli  the  beam  is  supported  ;  and  if  the  weights  at  the 
two  ends  change  places,  what  weiglit  must  be  added  to  the 
lighter  to  jireserve  ecjuilibrium  ? 

Ahs.  12  ins.  from  one  end  ;  27  lbs. 

23.  Two  forces  of  4  lbs.  and  8  lbs.  act  at  the  ends  of  a 
bar  18  ins.  long  and  make  angles  of  120"  and  !)0  with  it; 
tlnd  the  ])oint  in  the  l)ar  at  which  the  resultant  acts. 

Ani*.  fl  (4  —  1/3)  ins.  from  the  4  lbs.  end. 

24.  A  weight  of  24  lbs.  is  Kusi)eiided  by  two  flexible 
strings,  one  of  which  is  horizontal,  and  tiie  othei  is  inclined 
at  an  angle  of  30"  to  the  vertical.  What  is  the  tension  in 
each  string  ?  Ahh.  8  \/3  lbs. ;  Ki  ^3  lbs. 

2r).  A  pole  12  ft.  long,  weighing  25  lbs.,  rests  with  one 
end  against  the  foot  of  a  wall,  and  from  a  point  2  ft.  from 
the  other  end  a  cord  runs  horizontally  to  a  point  in  the 
wall  8  ft.  from  the  ground  ;  find  the  tension  of  the  cord  and 
the  pressure  of  the  lower  end  of  the  jiole. 

Ans.  11.25  lbs.;  27.4  lbs. 

26.  A  body  weighing  G  lbs,  is  placed  on  a  smooth  i>lane 
wiiich  is  inclined  at  30°  to  tlie  horizon  ;  find  the  two  direc- 
tions in  which  a  force  e(|nal  to  the  body  may  act  to  produce 
ecpiiiibrium.  Also  find  what  is  the  pressure  on  the  plane 
in  each  case. 

Ans.  A  force  at  00"  with  the  jilane,  or  vertically  upwards  ; 

It  =  0  \/3,  or  0. 

27.  A  rod.  AB,  5  ft.  long,  without  weight,  is  hung  from 
a  point,  C,  by   two  strings  which   are  attached  to  its  end.! 


liMi 


94 


EXAMPLES. 


and  to  the  point ;  the  string,  AC,  is  3  ft.,  and  BC  is  4  ft.  in 
Icngtii,  and  a  weight  of  a  ll)s.  i,s  hung  from  A,  imd  a  weight 
of  ;j  lbs.  from  li  ;  find  the  tensions  of  tiie  strings. 

Ans.   VS  lbs.;  2  VS  lbs. 

28.  Find  tlie  height  of  a  cylinder,  which  can  just  rest  oi: 
an  inclined  \^\^\m,  the  angle  of  which  is  00°,  the  diamctei 
of  the  cylinder  being  G  ins.  and  its  weiglit  acting  at  the 
middle  point  of  its  axis.  Ans.  3.4G  ins. 

29.  Two  equal  weights,  /',  Q,  are  connected  by  a  string 
wiiich  passes  over  two  smooth  pcg.s,  J,  B,  situated  in  a 
horizontal  line,  and  supports  a  weight,  W,  which  hangs 
from  a  smooth  ring  through  which  tlie  string  passes;  find 
the  position  of  equilibrium. 

Ans.  The  depth  of  the  ring  below  the  lino 

W 


AJJ  = 


AB. 


30.  Tlie  resultant  of  two  forces,  P,  Q,  acting  at  an  angle, 
0,  is  =  (2w  +  1)  V/'^T~^;  when  they  act  at  an  angle, 

^  -  0,  it  is  =  (2?/t  —  1)  V'l'^  +  Qi ;  show  that  tan  0  = 

m  —  1 
m  +1' 

31.  A  uniform  heavy  beam,  AB  =  2a, 
rests  on  a  smooth  peg,  P,  and  against  a 
smooth  vertical  wall,  AD;  the  horizontal 
distiince  of  the  peg  from  the  wall  being 
h ;  find  the  incliniition,  0,  of  the  beam  to 
the  vertical,  and  the  pressures,  A' and  .9,  on  the  wall  and  jieg. 

.„..«  =  .„-.(;;)S.s.=  „(;;)'.  «  =  „^L-^. 

32.  Two  ('((ual  smooth  cylinders  rest    in  contact  on  two 
smooth  })lanes  inclined  at  angles,  «c  and  ti,  to  the  horizon; 


I  BC  is  4  ft.  in 

,  mid  a  weight 
ngs. 

. ;  2  VS  lbs. 

m  just  rest  or 
the  diametei 
acting  at  tln' 

IS.  3.4G  ins. 

d  by  a  string 
situated  in  a 
wliicli  hangs 

I  passes;  find 


g  at  an  angle, 
at  an  angle, 

that  tan  0  = 


wall  and  jiog. 

Va"  -  h'^ 

hi       ■ 

ntaet  on  two 
the  horizon; 


EXAMPLES. 


95 


find  the  inelination,  6,  tc  the  horizon  of  the  line  joining 
their  centres.  Atis.   tan  6=1  (cot  «  —  cot  /3). 

33.  A  beam,  5  ft.  long,  weighing  5  lbs.,  rests  on  a  ver- 
tical prop,  CD  =  2|  ft. ;  tlie  lower  end,  A,  is  on  a  liori- 
zontal  plane,  and  is  jtrevented  from  sliding  by  a  string, 
AD  =  3-J-  ft.;  find  the  tension  of  the  string. 

A  us.   r  =  f  I  lbs. 

34.  A  uniform  beam,  AB,  is  placed  with  one  end,  A, 
inside  a  sniootii  hemispherical  bowl,  with  a  point,  P,  rest- 
ing on  the  edge  of  the  bowl.  If  AB  =  3  times  the  radius 
n,  find  AP.  Ans.  AP  =  1.838  li. 

35.  A  body,  weight  W,  is  suspended  by  a  cord,  length  I, 
from  tile  jioint  A,  in  a  iiorizontal  plane,  and  is  tiirust  out 
of  its  vertical  position  by  a  rod  without  weiglit,  acting  at 
another  point,  B,  in  thi;  horizontal  iilane,  such  that 
AB  =  d,  and  making  the  angle,  0,  with  tlie  plane;  find 
the  tension,  7,  of  tiie  cord. 


Ans. 


T  =  W  -,  cot  b. 


Fig.35. 


3G.  Two  heavy  uniform  bars,  AB  and 
(X),  movable  in  a  vertical  plane  about 
their  extremities.  A,  D,  wiiicli  rest  on  a 
horizontal  plane  and  are  prevented  from 
sliding  on  it ;  find  their  position  of 
equilibrium  when  leaning  against  each 
other. 

Tjct  the  bars  rest  against  each  other  at  .  B,  and  let 
AD  =  a,  AB  =  b,  CD  =  c.  BD  =z  x,  W  and  If,  =  tiio 
weigiits  of  AB  and  CD,  respectively  acting  at  their  niiddlo 
points;  then  we  have 

'iA^W  {a'i  +  />!  _  a^)  _  HI',  (rtS  +  .r^  -  W)  (J«  +  x^  -  a% 

which  is  an  efpiatioii  of  the  fifth  degree,  and  hence  always 
lias  one  real  root,  the  value  of  which  may  bo  determined 
when  numbers  are  put  for  a,  b,  and  c. 


I* 


96 


EXAMPLES. 


37.  A  parabolic  curve  is 
placed  in  a  vortical  i)laiic  with 
its  axis  vortical  and  vertex 
downwards,  and  inside  of  it, 
and  against  a  peg  in  the  focus, 
and  against  the  concave  arc,  a 
smooth  uniform  and  heavy 
l)eam  rests  ;  required  the  posi- 
tion of  ecjuilibrium. 

Ijct  PB  be  the  beam,  of 
length  /,  and  of  weight  W, 
resting  on  the  peg  at  the  focus, 
F ;  let  AF  =  p  and  AFP  =  e. 


ng.36 
A>IS.    6  =  2  ons~l  i^'\^^ 


COS" 


(0' 


38.  Find  the  form  of  the  curve  in  a  vertical  i)lane  such 
that  a  heavy  bar  resting  on  its  concave  side  and  on  a  peg  at 
a  given  point,  say  the  origin,  may  be  at  rest  in  all 
l)ositions. 

Ans.  r  =  y  +  k  sec  0,  in  which  I  =r  tlie  length  of  the 
bar,  X-  an  arbitrary  constant,  and  0  the  inclination  of  the 
bar  to  the  vertical.  It  is  the  equation  of  the  conchoid  of 
Nicomedcs. 

39.  A  rod  wIkj.m  centre  of  gravity  is  not  its  middle  point 
is  hung  from  a  smooth  peg  by  means  of  a  string  attached 
to  its  extremities  ;  find  the  position  of  equilibrium. 

Atis.  There  are  two  iwsitions  in  which  the  rod  hangs 
vertically,  and  there  is  a  third  thus  defined  :— Let  F  be  the 
extremity  of  the  rod  remo(e  from  the  centre  of  gravity,  Ic 
the  distance  of  the  centre  of  gravity  from  the  middle  ])oint 
of  the  rod,  '^a  the  length  of  the  string,  and  2c  the  length  of 
the  rod  ;  then  measure  on  the  string  u  length    FI'  from  /' 

e(|uallo«(l  -f-     ),  and   place   the   point   /'  (.ver   the   peg. 

This  will  detine  a  third  position  of  equilibrium. 


Ffg.36 


=  2  COS" 


■(f)* 


•Heal  plane  such 
and  on  a  peg  at 
at    rest    in    all 

le  length  of  the 
iclination  of  tiic 
the  conchoid  of 


its  middle  point 
I  string  attached 
librium. 

1  the  rod  hangs 
—Let /'be  the 
re  of  gravity,  k 
ic  middle  ])oint 
2r  the  length  of 
ii    /•'/'  from  F 

over   the   peg. 

um. 


EXAMI'IjES. 


97 


40.  A  smooth  hcmispiiere  is  fixed  on  a  korizontal  plane, 
with  its  convex  side  turned  upwards  and  its  base  lying  in 
the  plane.  A  heavy  uniform  beam,  AB,  rests  against  tlie 
hemisphere,  its  extremity  A  lieing  just  out  of  contact  with 
the  horizontal  plane.  Supposing  that  A  is  attached  to  a 
rope  which,  ])assing  over  a  smooth  pulley  placed  vertically 
over  the  centre  of  the  hemisiihere,  sustains  a  weight,  find 
the  position  of  equilibrium  of  the  beam,  and  the  recpiisite 
magnitude  of  the  suspended  wcigiit. 

/l«.v.  TiCt  ir  be  the  weight  of  the  beam,  "la  its  length,  P 
the  suspeiuled  weight,  ;•  the  radius  of  the  hcmisi)here,  h 
tlie  height  of  the  i)ulley  above  the  plane,  0  and  0  the 
inclinations  of  the  beam  and  roi)e  to  the  horizon ;  then  the 
position  of  ecpiilibrium  is  defined  by  the  equations. 


r  cosec  0  =  A  cot  0, 
r  cosec^  0  =  a  (tan  (p  +  cot  0), 
which  give  the  single  equation  for  0, 

r  (r  —  a  sin  0  cos  0)  =  ah  sin^  0. 
sin  0 


Also 


p  =  \y 


w 


COS  (0  —  6) 
,^a  sin2  0  \/l^^l^\x^d 


(1) 
(2) 

(3) 
(4) 


41.  If,  in  the  last  example,  the  position  and  magnitude 
of  the  beam  be  given,  find  the  locus  of  the  pulley. 

Ahs.  a  right  line  joining  A  to  the  point  of  intersection 
of  the  reaction  of  the  bemisi)here  and  11'. 

4-*.  If,  in  the  same  example,  the  extremity.  A,  of  the 
beam  rest  against  the  plane,  state  how  the  nature  of  the 
proble  n  is  moditied,  and  find  I  lie  position  of  e(|uilibrium. 

Ans.  The  suspended  weight  must  be  given,  insti;ul  of 
being  a  ."csult  of  calculation.  Equation  (1)  still  holds,  but 
5 


^ 


'j8 


EXAMPLES. 


not  (2) ;  aud  the  position  of  equilibrium  is  defined  by  the 
equation 

Ph^  coss  0  =  War  sin'  0. 

43.  If  the  fixed  hcinispiierc  be  roi)hiced  by  a  fixed  splicre 
or  cylinder  resting  on  the  jdane,  and  the  extremity  of  the 
beam  rest  on  the  ground,  find  the  position  of  equilibrium. 

Ans.  If  h  denote  the  vertical  height  of  the  pulley  above 
the  point  of  contact  of  the  sphere  or  cylinder  with  the 
plane,  we  have 

n 

r  cot  r:  =  A  cot  <^, 


a 


6 


Pr  (1  +  cot  -  cot  0)  cos  0  =  Wa  cos  0. 

44.  One  end,  A,  of  a  heavy  uniform  beam  rests  against  a 
smooth  horizontal  plane,  and  the  other  end,  B,  rests  against 
a  smooth  inclined  plane  ;  a  rope  attached  to  B  passes  over 
a  smooth  pulley  situated  in  the  inclined  plane,  and  sustains 
a  given  weight;  find  the  position  of  equilibrium. 

Let  6  be  the  inclination  of  the  beam  to  the  horizon, «  the 
inclination  of  the  inclined  })lane,  W  the  weight  of  the  beam, 
and  P  the  suspended  weight  ;  then  the  position  of  equili- 
brium is  defined  by  the  equation 

cos  6  ( W  sin  «  —  2P)  =  0.  (1) 

Hence  we  draw  two  conclusiojis : — 

(rt)  If  the  given  quantities  satisfy  the  equation  If  sin  « 
—  2P  =  0,  the  beam  will  rest  in  all  positions. 

(b)  There  is  one  position  of  equilibrium,  namely,  that  in 
which  the  beam  is  vertical. 

This  position  requires  that  both  planes  he  conceived  as 
prolonged  ihrnugh  their  line  of  intorsoctinn. 

45.  A  uniform  beiim.  AB,  movable  in  a  vertical  plane 
about  a  smooth  horizontal  axis  fixed  at  one  extremity.  A,  is 


defined  by  the 


>y  a  fixed  sphere 
stremity  of  the 
)f  equilibrium. 
;he  pulley  above 
inder  with   the 


cos  d. 

I  rests  against  a 
B,  rests  against 
0  B  passes  over 
ne,  and  sustains 
ium. 

c  horizon, «  the 
?ht  of  the  beam, 
sition  of  equili- 

(1) 


[nation  W  sin  « 

lis. 

namely,  that  in 


be  conceived  as 

I  vertical  plane 
extremity,  A,  is 


EXAMPLES. 


99 


attached  by  means  of  a  rope  BC,  whose  weight  is  negligible, 
to  a  fixed  point  C  in  the  horizontal  line  through  A,  such 
tliiit   AB  =  AC;  find  the  pressure  on  the  axis. 

Ans.    U  6  =  <CAB,    IK  =  weight  of    beam,    the    re- 
action is 

_ 

iTTy  4sin2-  +  8ec2x. 


ito 


CHAPTER  IV. 

CENTRE  OF  GRAVITY*  (CENTRE  OF  MASS). 

66.  Centre  of  Gravity. — Gravity  is  the  name  given  to 
the  force  of  attraction  wliieh  the  cardi  exerts  on  all  bodies: 
tlie  ellects  of  this  force  arc  twofold,  (1)  statical  iu  virtue  of 
which  all  i)odics  exert  pressure,  and  (:i)  kinetical  in  virtue 
of  which  bodies  if  unsupported,  will  fall  to  tiie  ground 
(Art.  15).  The  force  of  gravity  vai'ies  slightly  from  place 
to  place  on  the  earth's  surface  (Art.  2'S)  ;  but  at  each  ])lace 
it  is  a  forci.'  exerted  ujwn  every  body  and  upon  every 
particle  of  the  hotly  in  directions  that  arc  normal  to  the 
earth's  surface,  and  which  therefore  converge  towards  the 
earth's  centre;  but  as  this  centre  is  very  distant  compared 
witli  the  distance  between  the  imrticles  of  any  body  of 
ordiiuiry  magnitude,  the  convergence  is  so  small  that  the 
lines  in  which  the  force  of  gravity  acts  are  sensiljly  parallel. 

The  centre  of  (jmrihi  of  a  bodif  is  the  poitif.  of  appJicntion 
of  the  re,sult(nit  of  alt  the  forces  of  (jrarity  vhieh  act  upon 
every  particle  of  the  body ;  and  since  these  forces  are 
practically  parallel,  the  problem  of  finding  its  position  may 
be  treated  in  the  mme  loay  as  that  of  finding  the  centre  of  a 
system  of  parallel  forces  (Arts.  45,  59,  (13).  The  centre  of 
gravity  may  also  be  defined  as  the  point  at  vhich  the  whole 
ireif/hf  of  a  body  acts.  Tf  the  body  l)e  supported  at  this 
point  it  will  rest  in  any  position  whatever. 

T//e  weiijht  of  a  i/ody  is  th-  resultant  of  all  the  forces  of 
i/rnri/y  irhich  act  upon  erery  particle  of  it,  and  is  equal  in 
iiiayniludr  and  dirrrlly  uppn^i/i'  /n  the  force  which  will  fast 
s'ippiirt   the   liiiily.      Since    the   centre'   of  gravity   is   here 


*  Callod  also  Cnitiv  c.f  ^f  sx  nii.I  Centre  of  InerHa ;  and  the  term  Ceniroid  hat, 
laiily  ('omo  into  use  lo  df.sij;iiato  lt.i ' 


OF    MASS). 

lie  name  given  to 
rts  on  all  bodies ; 
itical  in  virtue  of 
inotical  in  virtue 
1  to  tlie  ground 
glitiy  from  place 
but  at  each  ])lace 
and  upon  every 
:rc  normal  to  the 
erge  towards  tlic 
listant  compared 
of  any  body  of 
so  small  that  the 
sen8il)ly  parallel. 
int  of  applicatiuii 
!/  v'/iich  ad  upon 
these  forces  are 
its  position  may 
'/7  the  centre  of  a 
.  The  centre  of 
■  which  the  whole 
Hipported  at  this 

f  all  the  forces  of 
and  is  criual  in 

f  which  will  just 
gravity   is  here 

1  the  term  Centroid  hm 


CtSMTJlK    III'-   UUAVITY. 


101 


regarded  as  the  centre  of  parallel  forces,  it  is  more  truly 
coMceivcd  of  as  the  "'centre  of  mass;"  yet  in  dflerenee  to 
usage  we  shall  call  the  point  the  "centre  of  gravity." 

67.  Planes  of  Symmetry.  -Axes  of  Symmetry.— If 

a  homogejieous  body  be  syianielrical  with  reference  U>  any 
plane,  the  c(!ntre  of  gravity  is  in  that  i)lane. 

If  two  or  more  such  planes  of  symmetry  intersect  in 
one  line,  or  axis  of  symmetry,  the  centre  of  gravity  is  in 
that  axis. 

If  three  w  more  planes  of  symmetry  intersect  each  other 
in  a  point,  that  point  is  the  centre  of  gravity. 

By  obr;erving  these  principles  of  the  symmetry  of  the 
tigurc  there  are  many  eases  Mhere  the  centre  of  gravity  is 
known  at  once  ;  thus,  the  centre  of  gravity  of  a  straight 
line  is  its  midijle  point.  The  centre  of  gravity  of  a  circle 
or  of  its  circumference,  or  of  a  si)liere  or  of  its  suriaee,  is  its 
centre.  The  centre  of  gravity  of  a  parallelogram  or  of  its 
perimeter  is  the  point  in  which  the  diagonals  intersect. 
The  centre  of  gravity  of  a  cylinder  or  of  its  surface  is  the 
middle  of  its  axis  ;  and  in  a  similar  manner  we  shall 
frequently  conclude  from  the  symmetry  of  the  figure,  that 
the  centre  of  gravity  of  a  body  is  in  a  particular  line  which 
can  be  at  once  dt  .ermined. 

When  we  speak  of  the  centre  of  gravity  of  a  line,  we 
are  really  considering  a  material  line  of  the  same  density 
and  thickness  throughout,  whose  section  is  infinitesimal  ; 
and  when  we  consider  the  centre  of  gravity  of  any  surface, 
we  are  really  considering  the  surface  as  a  thin  uniform 
lamina,  the  "thickness  of  which,  being  uniform,  can  be 
neglected. 

68.  Body  Suspended  from  a  ■PoirA.— When  a  body  is 
suspended  from  a  point  about  which  if  can  tvrnfreeiyjt 
will  rest  with  its  centre  of  yravify  in  the  vertical  line  passinq 
throvgh  the  point  of  svspensinn.     For.  if  the  i)o:nt  of  sus- 


^m 


102  noDr  SUPI'OIiTED   ox   A   SVHFACE. 

pension  and  tlie  centre  of  gravity  are  not  in  a  vertical  line, 
llie  weight  acting  vertically  downwards  at  the  centre  of 
gnn  ity  ai'd  (ho  reaction  of  the  point  of  suspen.sion  verticaliv 
upwards  form  a  statical  couple  and  hence  there  will  lie 
rotation. 

69.  Body  Supported  on  a  Surface.—  When  a  body  is 
placed  un  a  surface  it  ivill  stand  ur  fall  accordiuff  as  the 
vertical  line  through  the  centre  of  gi-avity  falls  within  or 
without  the  base.  For  if  it  falls  within  the  base  the  reaction 
of  the  surface  upward  and  the  action  of  the  weight  down- 
ward will  be  in  the  same  vertical  line,  and  so  there  will  bo 
equilibrium.  But  if  it  falls  without  the  base  the  reaction 
of  the  snrface  upward  and  the  action  of  the  weight  down- 
ward form  a  statical  couple  and  hence  the  body  will  rotate 
and  fall. 

70.  Different  Kinds  of  Equilibrium.— According  to 

the  proposition  just  proved  (Art.  09)  a  body  ought  to  rest 
ujjon  a  single  point  without  falling,  provided  that  its  centre 
of  gravity  is  placed  in  the  vertical  line  through  the  point 
which  forms  its  base.  And,  in  fact,  a  body  so  situated 
would  be,  mathematically  speaking,  in  a  position  of  equili- 
brium, though  practically  the  equilibrium  would  not  sub- 
sist. The  body  would  be  moved  from  its  position  by  the 
least  force,  and  if  left  to  itself  it  would  depart  further  from 
it,  and  never  return  to  that  position  again.  This  kind  of 
equilibrium,  and  that  which  is  practically  possible,  are 
distinguished  by  the  names  of  nnstabU  and  stable.  Thus 
an  egg  on  either  end  is  in  a  position  of  unstable  equilibrium, 
but  when  resting  on  its  side  it  is  in  a  position  of  stable 
equiliLJum.  The  distinction  may  be  defined  generally  as 
follows : 

When  the  body  is  in  such  a  position  that  if  slightly  dis- 
placed it  tends  to  return  to  its  original  position,  the  equili- 
brium is  stable.     When  it  tends  to  move  further  away  from 


AVE. 


CE.\TRE    OF    OJiA  rTTV    OF   A     'miAXOLE. 


103 


1)  a  vertical  lino, 
it  the  centre  oT 
leusion  verticallv 
;e  there  will   lie 

-  When  a  body  is 
tccordiny  as  tlie 
falls  within  or 
base  the  reaction 
le  weight  down- 
so  there  will  bo 
;ise  the  reaction 
e  weight  down- 
body  will  rotate 

, — According  to 
ly  ought  to  rest 
d  that  its  centre 
ough  the  point 
ody  so  situated 
)sition  of  equili- 
woiild  not  sub- 
position  by  the 
irt  further  from 
.  This  kind  of 
ly  possible,  are 
d  s/able.  Thus 
^ble  equilibrium, 
jsition  of  stable 
led  generally  as 

it  if  slightly  dis- 
tion,  the  equili- 
thcr  away  from 


its  original  position,  its  equilibrium  is  uiistnhle.  When  it 
remains  in  /7s  ?<('»' ^m/Viow,  its  oquilil)ri urn  \»  neutral.  A 
.sphere  or  cylindrical  roller,  resting  on  a  liorizontal  surface, 
is  in  neutral  equilibrium.  In  stable  eqiiiUhrium  the  centre 
iif  (/rarity  occupies  tlie  lowest  possible  position;  and  in 
unstable  it  occupies  the  highest  position. 
We  shall  first  give  a  few  elementary  examples. 

71.  GJ.ven  the  Centres  of  Gravity  of  two  Masses, 
Ml  and  31 2,  to  find  the  Centre  of  Gravity  of  the  two 
Masses  as  one  System.— Let  g^,  denote  the  centre  of 
gr.ivity  of  the  mass  Mi.  and  g^  the  centre  of  gravity  of  the 
mass  1/g.     Join  g^  g^  and  divide  it  ai.  the  point,  0,  so  that 

^^  =  4^^ ,  then  O  is  the  centre  of  gravity  of  the  two 
Gg^       Ml 

masses  as  one  system  (Art.  45). 

72.  Given  the  Centre  of  Gravity  of  a  Body  of 
Mass,  M,  and  also  the  Centre  of  Gravity  of  a  part 
of  the  Body  of  Mass,  m,  to  find  the  Centre  of 
Gravity  of  the  remainder.— Let  0  denote  the  centre  of 
gravity  of  the  mass,  M,  and  gi  the  centre  of  gravity  of  the 
mass,  nil.  Join  Gdi  a^d  produce  it  through  0  tog^,  so  tliat 

^-h  —  _-J!ii — ,  then  (7o  is  the  centre  of  gravity  of  the 
Ggi       M-nii' 

remainder  (Art.  45). 

73.  Centre  of  Gravity  of  a  Triangular  Figure  of 
Uniform  Thickness  and  Density.— Let  ABC  be  the 

triangle;  bisect  BC  in  D,  and  join  AD; 

draw  any  line  bdc  parallel  to  BC  ;  then  it 

is  evident  that  this  line  will  be  bisected  by 

AD  in  d,  and  will  therefore  have  its  centre 

of  gravity  at  d;  similarly  every  line  in  the 

triangle  parallel  to  BC  will  have  its  centre 

of  gravity  in  AD,  and  therefore  the  centre  of  gravity  of  the 

triangle  must  be  somewhere  in  AD. 


m 


104 


CESTHE    OF   GHAVITY   OF   A     TRIANGLE. 


In  like  manner  tlic  centre  of  gravity  mnst  lie  on  tiic  lino 
BE  which  joins  B  to  the  miildie  point  (if  AC.  It  is  there- 
fore at  the  intersection,  G,  of  AD  an'i  BE. 

Join  I)K,  which  will  l)e  i>arallel  to  AB;  tiien  the  triangles, 
ABfl,  DE(/,  are  siniihir;  therefore 


AG 
Gl) 


AB 
1)E 


BC 
DC 


2 
1' 


or 


GD  =  |AG  =  |AD, 

Hence,  to  find  the  centre  of  (/rarity  of  a  triangle,  bisect  any 
side,  join  the  point  of  bisection  witli  tlie  ojiposite  angle,  the 
centre  of  yrarity  lies  one  third  I  he  way  vj)  this  bisection. 

Coil.  1. — If  tiiree  eq;.;.!  ])articles  be  placed  at  the  vertices 
of  the  triangle  ABC  their  centre  of  gravity  will  coincide 
with  that  of  the  triangle. 

For,  the  centre  of  gra\ity  of  the  two  eqnal  particles  at  B 
and  C  is  tlie  middle  \)o\ut  of  BC,  and  the  centre  of  gravity 
of  the  tliree  lies  on  tlie  line  joining  this  point  to  A. 
Similarly,  it  lies  on  the  line  joining  B  to  the  middle  of  AC. 
Therefore,  etc. 

CoH.  2. — The  centre  of  gravity  of  any  ])lune  polygon  may 
be  fonnd  by  dividing  it  into  triangles,  finding  the  centre  of 
gravity  of  each  triangle,  and  then  by  Art.  59  deducing  tlie 
centre  of  gravity  of  tlie  whole  ligure. 

Cor.  3. — Let  the  co-ordinates  of  A,  referred  to  any  axes, 
be  Xj,  ^1,  z,  ;  those  of  B.  .r^,  y^,  z^ ;  and  those  of  C,  .Tj, 
^3,  ^3 ;  then  (Art.  50).  the  co-ordinates,  r,  y,  «.  of  the  centre 
of  gravity  of  three  equal  particles  placed  at  A,  B,  C,  respec- 
tively, are 


3 


_  h 


,  y  -         3        » 

3      ^     ' 


ANGLE. 

it  lie  on  the  line 
V.    It  is  tliere- 

eii  the  triangles, 


angle,  bisect  any 
posite  angle,  the 
'iin  bisection. 

tl  at  tlie  vertices 
ty  will  coincide 

,1  particles  at  B 
eiitre  of  gravity 
na  point  to  A. 
e  middle  of  AC. 


ine  polygon  may 
ng  the  centre  of 
)9  deducing  the 

red  to  any  axes, 

those  of  C,  .Tg, 

,  z,  of  the  centre 

A,  B,  C,  respec- 


CENTRE  OP  ORAVTTY  OP  A   PYRAMID. 


106 


which  are  also  the  co-ordinates  of  the  centre  of  gravity  of 
the  triangle  ABC  (Cor.  1). 

74.  Centre  of  Gravity  of  a  Triangular  Pyramid  of 
Uniform  Density. — Let  D-ABC  be  a  triangulai- pyramid; 
bisect  AC  at  E;  join  BE,  DE;  take  EF 
=  ^EB,  then  F  is  the  centre  of  gravity  of 
ABC  (Art,  73).  Join  FD  ;  draw  ab,  be,  ca 
l)arallcl  to  AB,  BC,  CA  resi)ectivcly,  and 
let  DF  meet  the  plane,  abc,  at  /;  join  bf 
and  produce  it  to  meet  DE  at  e.  Then 
since  in  the  triangle  ADC,  ac  is  parallel 
to  AC,  and  DE  bisects  AC,  e  is  the  middle  point  of  ac\  also 


Fig.3S 


BF  "~  DF 


EF' 


but 
therefore 


EF  =-  iBF, 
ef=W\ 


therefore /is  the  centre  of  gravity  of  the  triangle  dbc  (Art. 
73).  Now  if  we  suppose  the  pyramid  to  be  divided  by 
planes  parallel  to  ABC  into  an  indefinitely  great  number  of 
triangular  lamina-,  each  of  these  lamiuiB  has  its  centre  of 
gravity  in  DF.  Hence  the  centre  of  gravity  of  the  pyramid 
is  in  DF. 

Again,  take  EH  =  ^ED ;  join  FB  cutting  DF  at  G. 
Then,  as  before  the  centre  of  the  pyramid  must  be  on  BH. 
It  is  therefore  at  the  intersection,  CI,  of  the  lines  DF 
and  BH. 

Join  FH  ;  then  FH  is  parallel  to  DB.  Also,  EF  =  ^EB, 
therefore  FH  =  ^DB ;  and  in  the  similar  triangles,  FGII 
and  BGD,  we  have 


FG 
DG 


FH 
DB 


1 

3' 


therefore 


FG  =  JDG  =  pF. 


106 


CES'TliK   OF  QUA  Vtfy  OF  A    CONE. 


Hence,  the  reiifre  of  gravity  of  the  pyramid  is  07ie-fonrth 
of  the  way  ii/i  the  line  joininf/  the  centre  of  gravity  of  the 
bane  with  the  vertex.  (Todluiiitcr's  Statics,  p.  108.  Als(» 
Pratt's  Meelianics,  j).  T);?.) 

Cou.  1  -Tlie  cent  re  of  gravity  of  four  equal  particles 
placed  at  the  vertices  of  the  i)yrainid  coincides  with  the 
centre  of  gravity  of  the  pyramid. 

Cor.  2.— Let  {x^,y.^,  zj  be  one  of  the  vertices ;  {x^,  y^,  z^) 
a  second  vertex,  and  so  on ;  let  (x,  y,  i)  be  the  centre  of 
gravity  of  the  pyramid  ;  then  (Art.  59) 

«  =  i  C^'l    +  *'2   +  -f  3   +  ^i)> 

«  =  i  (^1  +  ^2  +  23  +  24)- 

Cor.  3. — The  perpend icnlar  distance  of  the  centre  of 
gravity  of  a  triangular  pyramid  from  the  base  is  equal  to  } 
of  the  height  of  the  pyramid. 

75.  Centre  of  Gravity  of  a  Cone  of  Uniform 
Density  having  any  Plane  Base. — Consider  a  pyramid 
whose  base  is  a  polygon  of  any  number  of  sides.  Divide 
the  base  into  triangles;  join  the  vertex  of  the  i)yraniid  with 
the  vertices  of  all  the  triangles ;  then  we  may  consider  the 
l)yramid  as  composed  of  a  number  of  triangular  pyramids. 
Now  the  centre  of  gravity  of  each  of  the;^e  triangular 
pyramids  lies  in  a  plane  whose  distance  from  the  l)ase  is 
one-fourth  of  the  height  of  the  ))yramid  (Art.  74,  Cor.  li) ; 
therefore  the  centre  of  gravity  of  the  whole  pyramid  lies  in 
tills  plane,  /.  e.,  its  perpendicular  distance  from  tlio  base  is 
one-fourth  of  the  height  of  the  jjyraniid. 

Again,  if  we  suppose  the  ))yramid  to  be  divided  into  an 
indefinitely  great  number  of  lamiiue.  as  in  Art.  74,  each  of 
these  lamina)  has  its  centre  of  gravity  on    the  right  line 


ONE. 


CENTRI-:    OF    OHA  rlTY. 


lol 


lid  is  one-fourth 
of  gravifii  of  Ihr 
s,  p.  108,     Also 


'  equal   particles 
ncides  with  the 


tices;  {pi;^,y^,z^) 
je  tlie  centre  of 


), 


f  the  centre  of 
ase  is  equal  to  \ 

e  of  Uniform 

sider  a  pyratuicl 
)f  sides.  Divide 
he  i)ynuiiid  with 
nay  consider  the 
igular  pyramids. 
the:-;o  triangular 
from  the  Imse  is 
Art.  7-t,  Cor.  ;J) ; 
pyramid  lies  in 
from  the  base  is 

divided  into  an 
Art.  74,  each  of 
I    the  right  line 


joining  the  vertex  to  the  centre  of  gravity  of  the  base;  and 
hence  the  centre  of  gravity  of  the  whole  pyramid  lies  on 
this  line,  and  hence  it  must  be  one-iuurth  the  way  up  this 
line.  There  is  no  limit  to  the  number  of  sides  oi  the  j)()ly- 
gon  which  forms  the  base  of  the  pyramid,  and  hence  tiiey 
may  form  a  continuous  curve. 

Therefore,  the  centre  of  (/rariti/  of  a  cone  whose  base  is 
any  plane  curve  whatever  is  found  by  joinimj  tlie  centre  of 
(/ravity  of  the  base  to  tlie  vertex,  and  taking  a  point  one- 
fourth  of  the  way  vp  this  line. 

76.  Centre  of  Gravity  of  the  Frustum  of  a  Pyra- 

niid._Let  A\M^-abc  (Fig.  :)«)  l)e  the  frustum,  formed  by 
tiie  removal  of  the  pyramid,  \)-abc,  from  the  whole  pyramid, 
I)-ABt! ;  let  // 1  and  //  be  the  perpendicular  heights  of  these 
pyramids,  respectively;  let  m  and  3/ denote  their  masses; 
and  let  «,,  Zg,?  denote  the  perpendicular  distances  of  the 
centres  of  gravity  of  the  pyramids  D-ABC!,  and  \)-abc,  and 
the  frustum,  from  the  base ;  then  we  have  (Art.  59,  Sch.  1 ) 


or 


But 


ifz, 

= 

z{M- 

m)  + 

wizg; 

8 

~     M 

2,   = 

—  «J2g 

—  m 
H 

• 

4' 

(1) 


.3=(F. 


h,)+\  =  H-\K. 


Also,  the  masses  of  the  pyramids  are  to  each  other  as  thoir 
volumes*  by  (1)  of  Art.  10,  and  therefore  as  the  cubes  of 
their  heights,     llenco  (I)  becomes 

♦  If  tlH!  bodlcB  nre  tionioKcnuonH.  iho  volumoH  or  tho  wolRlitB  arc  proportional  to 
th«  :na«B«B,aiid  may  bu  »ubBl'.lul>il  f')r  iIhuii. 


ita 


108 


z  = 


EXAMPLES. 


4'  tP-hi^ 


H 


T 


(2) 


Instead  of  the  heights  we  may  use  any  two  corresponding 
lines  in  the  lower  and  upper  bases,  to  whicii  the  heights  are 
proportional,  as  for  example  AB  and  ab.  Denoting  these 
lines  by  a  and  b,  and  the  altitude  of  the  frustum  by  h,  (2) 
becomes 

h    a^  +  Ub  +  3J2 


z  = 


4         rt2  _|_  f,l)   ^   J2 


This  is  true  of  a  frustum  of  a  pyramid  on  any  base,  a 
and  b  being  homologous  sides  of  the  two  ends,  and  hence  it 
is  true  of  tiie  frustum  of  a  cone  standing  oti  any  plane  base. 


EXAMPLES. 

1,  Find  the  centre  of  gravity  of  a  tra])ezoid  in  terms  of 
the  lengths  of  the  two  parallel  sides,  «  and  b,  and  of  the 
line,  //,  joining  their  middle  points. 

Tttke  iiiomeuls  with  referoncc  to  the  l-mger  parallel  side. 

Alls.  On  the  line  bisecting  the  parallel  sides  and  at  a 

//    a  +  2b 


distance  from  its  lower  end 


;}     a  +  b 


2.  If  out  of  any  cone  a  similar  cone  is  cut  so  that  their 
axes  are  in  the  same  line  and  their  I)ase8  iu  the  same  plane, 
find  the  height  of  the  centre  of  gravity  of  the  remainder 
above  the  base. 

Tuki'  iiionientH  with  relereiu'c  to  tho  base. 


lyTEGRATIOX  FORMULA. 


109 


I-  3/^2 


(2) 


0  corresponding 

the  lioightsare 

Denoting  these 

istum  by  It,  {'Z) 

(3) 

on  any  base,  a 
ds,  and  hence  it 
i  any  plane  base. 


loid  in  terms  of 
id  b,  and  of  the 

Ik'l  side, 
sides  and  at  a 


it  so  that  their 

the  same  phme, 

f  the  remainder 


.In*'.   \'~ ?^*,  where  //.  is  the  height  of  the  original 

cune,  and  /;',  the  height  of  that  wiiich  is  cut  out  of  it. 

3.  If  out  of  any  cone  anotlier  eonc  is  cut  haviirg  tlio 
siiine  base  and  their  axes  in  tlie  same  Hne,  find  the  heiglit 
of  llie  centre  of  gravity  of  the  remainder  above  the  base. 

Alls.  }(/(4-//,),  where  //  and  li y  are  tlie  respective 
iieights  of  tiie  original  cone  and  the  one  that  is  cut  out 
ot  it. 

4.  If  out  of  any  right  cylinder  a  cone  is  cut  of  the  same 
base  and  height,  find  the  centre  of  gravity  of  the  remainder 

Ans.  |ths  of  tiie  height  above  the  base. 

77.    Investigations    Involving   Integration.  — Tlie 

general  formula'  for  the  co-ordinates  of  the  centre  of  gravity 
vary  according  as  we  considei-  a  material  line,  an  area  or 
thin  lamina,  or  a  solid  ;  and  assume  different  forms  accord- 
ing to  the  manner  in  wliieh  the  matter  is  supposed  to  be 
divided  into  infinitesimal  elements. 

In  either  case  the  principle  is  the  same;  the  quantity  of 
matter  is  divided  into  an  infinite  number  of  infinitesimal 
elements,  the  mass  of  the  element  being  dm  ;  multiplying 
the  element  by  its  co-ordinate,  x,  for  example,  we  get 
X  '  dm,  which  is  the  moment  of  the  element*  with  respect  to 
the  plane  tfz  (Art.  03) ;  and  ./'.*•  •  dm  is  the  sum  of  the 
moments  of  all  the  elements  with  respect  to  the  plane  yz, 
and  which  corresponds  to  X/'a:  of  Art.  03..  Also,  /dm  is 
the  sum  of  the  masses  of  all  the  elements  which  correspond 
to  i:/'  of  the  same  Article.  Hence,  dividing  tlie  former  by 
the  latter  we  have 


♦  Till'  inoinciil  iif  the  fDrce  iiiIIiil'  oii  clenn'iit  itni  I-  slrlotlv  ilni  r/.r,  hill  since 
llii' coii^miit  f/  aiipoiM!'  Ill  ImiIIi  li'i-cns  of  cxiiri'KHloii  for  i  oiiriliiiiilcw  cil' eriilri'  o' 
nmviiy,  it  limy  In'  iMiillti'd  and  II  hcroniCK  iiKirr  coiivciiicnl  lo  npoak  dftlic  momfiil 
(if  the  ilewent,  iiicaiilii).'  by  11  Ilic  pnidiut  nf  llio  m\\<-»  of  Ilic  cloniciil  itm.  mid  Iih 
urni,  r  i'lic  iiiiiiiiriil  ol'uu  climienl  iiiuttHuruu  Itu  I'ffcQt  In  deturuiiulBg  tliu  iioeitliii: 
of  ilu' guutro  of  gravity. 


itaM 


110  CENTRE   OF  GRAVITY  OF  A    LINE. 

-  _  J'x  '  dm 

~    J'dm 


(1) 


Similarly  y  =  ll^^,  (2) 


fz  •  dm  ^ 
TdnT' 


z  =  -^- :  (3) 


the  limits  of  integration  being  determined  by  the  form  of 
the  l)ody  ;  the  sign,  ./',  is  used  as  a  general  symbol  of  sum- 
mation, to  be  rej)laced  by  the  symbols  of  single,  double,  or 
triple  integration,  according  as  dm  denotes  the  mass  of  an 
elementary  length  or  surface  or  solid.  Hence,  the  co-or- 
dinate of  the  centre  of  gravity  referred  to  any  plane  is  equal 
to  the  sum  of  the  moments  of  the  elemetits  of  the  mass 
referred  to  the  same  plane  divided  by  the  sum  of  the  elements, 
or  the  li'hole  mass.  If  the  body  has  a  plane  of  symmetry 
(Art.  67),  we  may  take  it  to  be  the  plane  xy,  and  only  (1) 
and  (2)  are  necessary.  If  it  has  an  axis  of  symmetry  wo 
may  take  it  to  be  the  axis  of  x,  and  only  (1)  is  necessary. 

78.  Centre  of  Gravity  of  the  Arc  of  a  Curve.— If 

the  body  whose  centre  of  gravity  we  want  is  a  material  line 
in  the  form  of  the  arc  of  any  curve,  dm  denotes  the  mass  of 
an  elementary  length  of  the  curve. 

Let  ds  =  the  length  of  an  element  of  the  curve ;  let 
h  =■  the  area  of  a  normal  section  of  the  curve  at  the  point 
(x,  y,  z),  and  let  p  =  the  density  of  tiie  matter  at  this 
point.  Then  (Art.  11),  we  have  dm  =  kpds,  which  is  the 
mass  of  the  element  ;  multiplying  this  mass  by  its  co-or- 
dinate, a-,  for  example,  we  have  the  moment  of  the  element, 
(kp.rds),  with  respect  to  the  plane,  yz. 

ITenee,  sul)stitntiiig  for  dm  in  (1),  (2),  (3).  of  Art.  77. 
ilie  linear  element,  kpds,  we  obtain,  for  the  jiosition  of  the 
centre  of  gravity  of  a  body  in  the  form  of  any  curve,  the 
equations 


\'E. 


EXAMPLES. 


Ill 


(1) 
(2) 
(3) 

iy  the  form  of 
(Tnbol  of  sutn- 
gle,  double,  or 
lie  muss  of  an 
?ncc,  the  co-or- 
'  plane  is  equal 
's  of  the  mass 
of  the  elements, 
!  of  .symmetry 
,  and  only  (1) 
symmetry  we 
is  necessary. 

a  Curve.— If 

X  material  line 
tes  the  mass  of 

;he  curve ;  let 
0  at  the  point 
matter  at  thid 
',  which  is  the 
!a  by  its  co-or- 
)f  till'  clement, 

3).  of  Art. ::. 

)osition  of  the 
any  curve,  the 


X  = 


y  = 


e  = 


fkpxds 
J'kpda  ' 

fkpyds 
fkpds  * 

fkpzds 
fkpds 


(1) 

(2) 
(3) 


The  quantities  ^*  and  p  must  be  given  as  functions  of  the 
position  of  the  point  {x,  y,  z)  before  the  integrations  can 
be  performed. 

If  the  curve  is  of  double  curvature  all  three  equations 
aie  required.  If  it  is  a  plane  curve,  we  may  take  it  to  be 
in  the  plane  xy,  and  (1)  and  (2)  are  suflfteicnt  to  determine 
the  centre  of  gravity,  since  i  =  0.  If  the  curve  has  an  axis 
of  symmetry,  the  axis  of  x  may  be  made  to  coincide  with 
it,  and  (1)  is  sufficient. 

EXAMPLES. 

1.  To  find  the  centre  of  gravity  of  a  circular  arc  of  uni- 
form thickness  and  density. 

Let  BO  be  the  arc,  A  its  middle  point, 
and  0  the  centre  of  the  circle.  Then  as 
the  arc  is  symmetrical  with  respect  to  OA 
its  centre  of  gravity  must  lie  on  this  line. 
Take  the  origin  at  0,  and  OA  as  axis  of  a;. 
Then,  since  k  and  p  are  constant,  (1)  be- 
comes 


_  fxds 
"  J'ds' 


(1) 


r  being  the  co-ordinate  of  any  point,  P,  in  the  arc.  Lei  6 
be  liie  angle  I'OA.  and  a  the  radius  of  the  circle,  and  let 
,1  —  the  angle  BOA.     Then 


* 


112 


and 


EXAMPLES. 

X  =  a  cos  6, 
ds  =  a  dd. 


Hence  x  = 


fj 


:'  COS  e  dd 


fj 


=  a 


/: 


cos  d  do 


dd 


If 


=  a 


sina 


a 


Therefore,  the  distance  of  the  centre  of  gravity  of  the  arc  of 
a  circle  from  the  centre  is  the  prodnct  of  the  radius  and  the 
chord  of  the  arc  divided  by  the  leMjth  of  the  arc. 

Cor. — The  distance  of  the  centre  of  gravity  of  a  semi- 

2a 
circle  from  the  centre  is  — • 

It 

2.  Find  the  centre  of  gravity  of  the  quadrant,  AD,  (Fig. 
39),  referred  to  the  co-ordinate  axes  OX,  OY. 
The  equation  of  tlie  circle  is 

x^  -^  y'^  =  a\ 


and 


dx  _    dy   _  Vdx^  +  dy^  _  ds  ^ 

7  ~"  ^^  ~  ^w+^  ~  ^  * 

,         axdx 
.*.    zds  = , 

yds  =  adx, 

J        adx 
ds  = : 

y 


which  in  (1)  and  (3),  after  canceling  k  and  p,  give 
C**     X  da- 


rn = 


^^0   VoT-a^       l-<"'-^)U       2a 


I 


dx 


0  Va*  —  «' 


b^"-'li 


=  a 


8\na 


Ity  of  the  arc  of 
radius  and  the 


arc. 


•avity  of  a  senii- 


laiit,  AD,  (Fig. 
Y. 


ds 


P,  give 

T^)^  I" 


2a 


EXAMPLES. 


113 


J^'^"^ 


w: 


I 


dx 


0  v^a^  _  x^ 


L-'-^I 


2a 


3.  Find  the  centre  of  gravity  of  the  arc  of  a  cycloid. 

Take  the  origin  at  the  starting  point  of  the  cycloid,  and 
lot  the  base  be  taken  as  the  axis  of  x.  The  equation  of 
the  curve  is 


X 


=  a  vers~>  -  -  —  {2ay  —  y^)^  ; 


dx dy        _     ds 

y*~(2«-#~(2«)*' 

it  is  evident  that  the  centre  of  gravity  will  be  in  the  axis  of 
the  cycloid  ;  therefore  1  —  mi ;  and  as  k  and  p  are  constant, 
(2)  becomes 


''o    Tia  — 


( 


!f)' 


dy 


^. 


'o    (2«  —  y)^ 
Cob. — For  the  arc  of  a  semi-cycloid,  we  get 
5  =  \a,    y  =  |rt. 

4.  Find  the  centre  of  gravity  of  a  circular  arc  of  uniform 
section,  the  density  varying  as  the  length  of  the  arc  from 
one  extremity. 

Let  AH  (Fig.  ;U)),  be  the  arc  :  let  ji.  be  the  density  at  the 
nnits  distance  from  A.  tiien  its  will  be  the  density  at  the 
distance  .v  from  A  ;  let  0.\  be  the  axis  of  r,  and  tt  tiie 
Z_  AOB.  Then,  putting  iis  for  p.  and  a  cos  0,  a  sin  d,  a  dO, 
and  ad,  for  x,  y,  ds,  and  *•,  in  (1)  and  (2), 


^ 


114 


EXAMPLES. 


T 


I  k  '  \i,  ad  •  a  cos  6  ■  n  dO  IS  ccis  0 

j,  _  1 „'[o 

J  k  •  fiad  •  a  (16 


lid 


=  n- 


J 


ddd 


=  2a 


«  sin  «  +  cos  «  —  1 


/  k  •  uaO  •  a  sin  6  .  a  do  10 sin  0 

7.  -  'i yo 


dd 


J  k  •  fiad  •  a 


dd 


0de 

t/O 


=  2a 


sin  «  —  «  cos  « 

^ 


CoK. — For  a  quadrant  we  get 


4a 


8a 


5.  Find  the  centre  of  gravity  of  one-lialf  ot  a  loop  of  a 
/eniiiiscate  whose  equation  is  r*  =:  i^  cos  20,  I  being  the 
length  of  the  half-looii. 


Here 


dr 


rdfl 


da 


—a?  sin  29      a'  cos  26      a» '  " 


etc. 


Ans,  X  z= 


2^1' 


y  =  a' 


,2*  —  1 


6.  Find  the  centre  of  gravity  of  a  straight  rod,  the  den- 
sity of  which  varies  as  the  wth  power  of  the  distance  of 
each  point  from  one  end. 

Take  the  orifj^in  iit  this  end,  su])i)08e  tlie  axis  of  .v  to  coincide  witli 
tlie  axis  uf  tbu  rod,  and  let  I  ~  the  lengtli  uf  tlie  rod. 


Ans.  X  =  — —-^  I. 
n  +  2 


CCS  d  lie 


eeie 


sin  d  dd 


Odd 


] 


t  a  loop  of  a 
',  I  being  the 


rod,  the  den- 
le  distance  of 


to  cninciilp  witli 


n  +  1 


I. 


cEyriiE  OF  iiUAvny  or  a.v  area. 


115 


T.  Find  tlie  centre  of  gravity  of  the  arc  of  a  semi-car- 
dioid,  its  eiiuation  being 

r  =  a  (1  +  cos  6), 

Ans.  The  co-ordinates  of  tlie  centre  of  gravity  referred 
to  the  axis  of  tlie  curve  and  a  perpendicular  through  the 
cusp,  as  axes  of  x  and  y,  are 

i  =  y  =z  \a. 


79.  Centre  of  Gravity  of  a  Plane 
^rea. — Let  AHCD  i)e  an  area  bounded 
by  the  ordinates,  AC  and  liD,  the  curve 
AB  whose  e(puition  is  given,  and  the  axis 
of  X  ;  it  is  recpiired  to  find  the  centre  of 
gravity  of  tliis  area,  the  lamina  (Art.  G7) 


We 


being  supposed  of  uniform  thickness  and  density, 
divide  the  area  into  an  infinite  number  of  infinitesimal 
elements  (Art.  77).  Suppose  this  to  be  done  by  drawing 
ordinates  to  the  curve.  Let  PM  and  QN  be  two  consecu- 
tive ordinates,  let  {x,  y)  be  the  point,  P,  and  let  g  be  tlie 
centre  of  gravity  of  the  trapezoid.  MPQN,  whose  breadth  is 
dx  and  whose  [larallel  sides  are  y  and  y  +  dy.  The  area  of 
til  is  trapezoid  is  y  dx,  (Cal.,  Art.  184). 

Let  p  be  the  density  and  ^- the  thickness  of  the  lamn.a. 
Then  (Art.  11)  we  have  dm  =  kfty  dx,  w]v.(:h  is  the  mass 
of  the  element  MPQX;  multiplying  this  mass  by  its  co-or- 
di suite,  X,  for  example,  we  have  the  moment  of  the  element 
(k-rxydx),  with  respect  to  OY,  and  niultii)lying  by  the 
other  co-ordinate,  |//,  we  have  tlit-  moment  with  respect  to 
OX.  Hence,  substituting  for  dm  in  (1)  and  {i)  of  Art.  77, 
till'  surface  elenu'iit.  kpi/  dx.  and  remembering  that  k  and  p 
;mv  constants,  we  ol)taiii,  fur  the  ])ositioii  of  the  centre  of 
gravity  of  a  body  in  the  form  of  a  plane  area,  the  eciuations, 


116  EXAMVLES. 

the  integrations  extending  over  the  whole  area  CABD. 


EXAMPLES. 

1,  Find  the  centre  of  gravity  of  tlie  area  of  a  semi-parab. 
ola  whose  equation  is  //^  =  'ipx. 

Let  a  =  the  axis,  and  b  tiie  extreme  ordinate,  then  we 
have  from  (1) 

/    V'-ip  x^'  dx         I   x^  dx 

-  _  ^^0 -A) 

"—     /K.     _  —      ,^  —  t«» 

/    V'ipx^dx        I   xi  dx 

Ipxdx  I-    I   X 


dx 


I    <.pxax  ,-    I    xax 

/    V^px^dx         ^  x^  dx 

2.  Find  the  centre  of  gravity  of  the  area  of  an  elliptic 
quadrant  whoso  equation  is 


y  =  -  Va*  —  sfi. 


a 


Here       i 


Hxy  dx        f  *  (rt2  _  a;2)i  x  dx 


4a 


*  =  3^* 


(1) 


ea  CABD. 


f  a  semi-parab. 
linate,  then  we 


xdx 


I*. 


x^  dx 

a  of  an  elliptic 


)^  xdx 
^)^dx 


a?)dx 
7?)^dx 


EXAMPLES. 


-      u 


117 


Hence  for  the  centre  of  gravity  of  the  area  of  a  circular 
quadrant  we  havo 

4a 


X  =  ]f  =z 


3n 


.*].  Find  the  centre  of  gravity  of  the  area  of  a  semi- 
cycloid. 

Take  the  axis  of  the  cnrvo  as  axis  of  a;  and  a  tangent  at 
the  higliest  point  as  axis  of  y  ;  then  the  e<iuation  is  (Anal. 
Geom.,  Art.  157), 


,« 


y  =z  a  vers"'  -  +  V'^iax  —  x^; 

where  a  is  the  radius  of  the  generating  circle.    From  (1)  we 

have 

/^2o  rya^         p^      -fa 

_  ^  J^xydx  ^  [V  -  ,/  3  dyj^ 

^  \[y:<^-fx{^ax-x^)^dx^  ^     ^^  ^^^y  _  ^^3 
\yx  —  y{2ax  —  a^)^  dxT" 

since  when  a;  =  0  and  2a,  y  =  0  and  no, 
.«.    i  =  ^a. 
Also, 

f^y'  dx       [y^x  -  2  fyx  dy]^ 


7Ta  •  2a  —  ^Tra* 


118 


POLAR  EIjEMK.\TS   of  A    PLANE  AREA. 


y^x—%   I  y  {2ax  —  x^)^  dx  \~ 


Snu^ 


\fx—2aj  (2rtx— x2)*  \m-^  "  dx—2j^{2(ix-x^)  ,/xX' 


Srra^ 


[y'x  —  2(ix^  +  p-^  —  2aJ{2ax  —  x^)^  vers-i  -  dxX 


3     2   3  ^«' 


wliicli  tlie  student  can  verify  by  ussuniing 


vers"'--  =  6, 
a 

(See  Todlmnter's  Statics,  p.  118.) 

80.  Polar  Elements  of  a  Plane 
Area. — Let  Ali  l)e  the  arc  of  a  curve, 
and  let  it  be  '-Hiuirod  to  liiid  the  centre  of 
ffi-avity  of  llie  area  bounded  l)y  the  arc 
AH  and  tiie  oxtrcnie  radii-voctors,  OA 
and  OB,  drawn  from  the  pole,  0,  to  the 
cxtrt'initios  of  tlie  arc. 

Divide  tiio  area  into  infinitesimal  trian},des,  such  as  POQ, 
inchidcd  between  two  consecutive  riidii-vcclors,  01'  and 
OQ.  liCt  (r.  0)  l)c  tiie  point,  /'.  tlicn  tlie  area  of  the 
(■lenient.  POQ  =  \i^ dO  (Cal.,  Art.  l!)l)  ;  aad  if  the  thick- 
ness and  density  of  tlio  lumiua  are  uniform,  the  centre  of 


ng.4i 


IREA. 


2ax-.r^)  (h-l^ 


^       -]2a 

a     Jo 


,  such  as  POQ. 
Dtortj,  OP  iiiid 
10  iirou  of  I- he 
1  if  tlio  thick- 
Ihu  oi'iitro  of 


KX AMPLE. 


119 


gravity  of  this  elementary  triangle  will  be  on  a  straight  lino 
drawn  from  0  to  the  middle  of  PQ,  and  at  a  distance  of 
two-thirds  of  this  straight  line  from  0  (Art.  73).  Hence 
the  co-ordinates  of  the  centre  of  gravity,  <j,  of  POQ,  are 
OM  and.  M^,  or, 

\r  cos  6,    and    \r  siu  0. 


Hence,  (Art,  77), 


/•)-3  cos  e  de 


_  _  ,r\r^\Oj^dO 


•'^dO 

/■/•3  sin  edO 
Ji^dO      '■ 


(1) 
(2) 


the  integrations  extending  over  the  whole  area,  x\OB. 


E  X  A  M  P  1.  E  , 

Find  the  centre  of  gravity  of  the  area  of  a  loop  of  Bcr- 
nouiili's  Lemniscate  wliose  e(|nation  is  /^  =  d^  cos  W. 

As  the  axis  of  the  loop  is  synunetrical  with  respect  to 
the  axis  oi  x,  y  —  0,  and  tlie  abscissa  of  the  centre  of 
gravity  of  the  whole  loop  is  evidently  tli.  same  as  that  of 
tiic  half-loop  abov  the  axis.  Substituting  in  (1)  f  )r  r  its 
value  a  cos*  W,  we  have 


S-U 


w 

A'osi  20  COS  e  dd 

«'n 


cos  20  dd 


tr 

_  A,,  f\\  _  2  sin"  0)'  d  sin  B. 


Put  sin  0  =    -;--,  then 


ita 


120 


UOVULE  ISTEaUA TIOX. 


Art  /*2  J.//  TT 

=  —~  /  c()s<  0  #  =  -     '-  •  I  s  (Cal.,  Art.  15T). 


TTa 


4V2 


81.  Double  Integration. — Polar  Formulae.— When 

the  density  of  tlio  lamina  varies  from  poinc  to  point,  it  may 
be  necessary  to  divide  it  into  elements  of  the  second  order 
instead  of  reetangnlar  or  triangular  elements  of  tlie  first 
order  (Arts.  79  and  80). 

Suppose  that  the  density  of  the  lamina  AOB  (Fig.  41), 
is  not  uniform.  If  we  divide  it  into  triangular  elements, 
POQ,  the  element  of  mass  will  be  no  longer  proj)ortional  to 
the  element  of  area,  POQ  =  ^I'^dO',  nor  will  the  centre  of 
gravity  of  the  triangle,  POQ,  be  fr  distant  from  0. 

Let  a  series  of  circles  be  described  with  0  as  a  centre, 
the  distance  between  any  two  successive  circles  being  dr. 
These  circles  will  divide  the  triangle,  POQ,  into  an  infinite 
number  of  rectangular  elements,  abed  =  rdOdr.  If  I'  is 
the  thickness  and  p  is  the  density  of  the  lamina  at  this  ele- 
ment, the  element  of  nuvss  will  he  dm  =  kprdddr\  and 
the  co-ordinates  of  its  centre  of  gravity  will  be  r  cos  0  and 
r  sin  0.     Hence,  from  (1)  and  {'I)  of  Art.  77,  we  have 


X   = 


and 


/'  /'/•  pr  cos  0  rdO  dr         f  j'kpi^cos  0 dO dr 
f  I  'kpr  do  dr  J  *  /  'kpr  dO  dr 


V  = 


/  /  kpr'  sin  0  dd  dr 

77A"- 


do  dr 


(1) 


(2) 


In  each  of  these  integrals  the  values  of  k  and  {>  are  (o  lie 
substituted  in  terms  of /•  and  ''.and  the  inlegralioiis  taken 
between  [iropcr  limits. 


(Cal.,  Art.  15T). 


irmulsB.— When 

to  point,  it  may 
he  second  order 
ents  of  tlie  tir-st 

AOB  (Fig.  41), 
iigular  elements, 
f  proportional  to 
ill  the  centre  of 
from  0. 

Ii  O  as  a  centre, 
'ircles  being  dr. 

into  an  infinite 
rdOdr.  If  k  is 
inina  at  this  ele- 

k  pr  dd  dr  ;  and 
II  be  r  cos  0  and 
7,  we  have 

los  0  do  dr 
--     —  ;      (1) 
r  de  dr 


(2) 


and  {>  arc  (o  be 
egralions  taken 


EXAMPLE. 


EXAMPLE. 


121 


Find  the  centre  of  gravity  of  the  area  of  a  cardioid  in 
which  the  density  at  a  point  increases  directly  as  its  distance 

from  the  cusp. 

Let  ju  =  the  density  at  the  unit's  distance  from  tiie 
cusp,  then  p  =  /*»•,  is  the  density  at  the  distance  r  from 

tiie  cusp. 

As  the  axis  of  the  curve  is  an  axis  of  symmetry  (Art.  GT), 
I  —  0,  and  the  abscissa  of  the  whole  curve  is  the  same  as 
for  the  half  above  the  axis  ;  then  (1)  becomes 


js  cos  0  f/0  dr 


w  = 


rr 

''O    ''0 

/     /   r'^dedr 

t'O    ^0 

/*V  cos  e  dO 

• 'n 


/  i^de 

by  performing  the  r-integration. 
The  etiuation  of  the  curve  is 

e 

r  =  o  (1  +  COS  0)  =  "id  cos*  ^' 

Substituting  this  value  for  r,  and  putting  ^j  =  0,  wo  have 

/*'cos«  0  {'i  eos»  ip  —  \)  d<t> 
•^0 


5  =  fa 


—  1^' 


/    cos*  tft  dip 


rita 


1^2  ItECTAMlULAR    FOKMUTjAE. 

82.  Double  Integration.— Rectangular  Formulse.— 

Lot.  a  series  of  cousocutive  sti'fiiglit  lines  bo  drawn  parallel 
to  tlie  axes  of  x  and  y  respectively,  dividing  the  area,  ABCD, 
(Fig.  40),  into  an  infinite  number  of  rectangular  elements 
of  the  second  order.  Then  the  area  of  each  element,  as 
abed,  =  ilx  dy ;  and  if  k  and  p  are  the  thickness  and  dcnsit.v 
of  the  lamina  at  this  element,  the  clement  of  mass  will  l)e 
dm  =  kp  dx  dy ;  and  the  co-ordinates  of  its  centre  of  gravity 
will  be  X  and  y.  Hence  from  (1)  and  (2)  of  Art.  77,  wo 
have 

J  J  k  px  dx  dy 

•=-7.> ;  (1) 

y   /  kp  dx  dy 

f  Jk  py  dx  dy 
I  I  kp  dx  dy 

the  integrations  being  taken  between  proper  limits. 

EXAMPLE 

Find  the  centre  of  gravity  of  the  area  of  a  cycloid  the 
density  of  which  varies  us  the  «th  power  of  the  distance 
from  the  base. 

Take  the  base  as  the  axis  of  .r  and  the  starting  point  as 
the  origin.    Then  tiie  o(iuation  of  the  curve  is 

X  —  a  vers'  ■    —  ['lay  —  y"^)^ ; 


y/'iay-f 


(8) 


ir  FormulaB.— 

B  drawn  parallel 
he  area,  ABCD, 
igular  oleinonts 
ach  element,  as 
ness  and  density 
)f  mass  will  l)e 
icntre  of  gravity 
)  of  Art.  77,  wo 


(1) 


(3) 


limits. 


a  cycloid  the 
)f  the  distance 

irting  point  as 
is 


SURFACE    OF    RKVOLUTIOX.  123 

Tx!t  p  —  /'//"  =  density  at  the  distance  y  from  the  base. 
It  is  evident  that  the  centre  of  gravity  will  be  in  the  axis  of 
tiie  cycloid  ;  therefore  x  =  ^^a  \  and  as  k   is  constant  (^) 


becomes 


/       /    ,f^Ulydx 

t/Q         *^0 

/       /   ydydx 
/       ?/«  +  8  dz 

n_+J  t/o 

nfla.       yn^^du 


r 

~  n  +  -i    r^     y"*^  dy^ 

_  «  +  1    2?t  +  5     ''^ ^/2ay  —  f . 

"^0        V'irty  —  ^^ 


y  = 


n  +  1    2ft  4-  5 
jT-f^ '  T+  3  ' 


83.  Centre  of  Gravity  of  a  Surface  of  Revolu- 
tion. -Let  a  surface  be  generated  l)y  Hie  revolution  of  the 
curve,  AH  (Fig.  4(1).  round  the  axis  of  .r.  Tlien  I  lie 
fleinentarv  arc.'  /'(,).  (==  */.<*).  generutes  an  element  of  tiu' 
surface  whose  area  ==  'i-^y  ds  (Cal..  Art.  VX\).  If  /is  Hi- 
lliiekne.><sand  p  the  density  of  tiie  lamina  or  siioll  m  tlii> 
element urv  zone,  the  element  of  mass  will  be  dm  =  2rthy  ds. 
Also  the  centre  of  gravity  of  this  «one  is  in  the  axis  of  .>■  ut 


^■ta 


124  EXAMPLES. 

the  point  Jf  wlioso  abscissa  is  x  and  ordiuate  0.    Hcnc*  (1) 
of  Art.  77  becomes,  af^or  cancelling  'Zir, 

I  kpxy  ds 

S  =  — (1) 

/  kpy  ds 

the  integrations  being  taken  between  proper  limits. 

EXAMPLES. 

1.  Find  the  centre  of  gravity  of  the  surface  formed  by 
tlie  revolution  of  a  semi-cycloid  round  its  base. 
The  equation  of  the  generating  curve  is 


X 


=■  a  vers-i  "  —  \/'iay  —  y^ ; 

.      (^ dg ds     ^ 

y        V'iay-f~  V2ay'' 

or  ds  =  ^^MM=. 

V'ia  —  y 

which  in  (1)  gives,  after  cancelling  V^  kp, 

r^_^ydy_ 
•A    Via  —  y 

•'o    y'i(,  —  If 

'i.  Find  the  centre  of  gravilyof  the  surface  formed  by 
the  revolution  of  a  semi-cycloid  round  its  axis. 

It  is  clear  that  thcccntn'of  gravity  lies  on  the  a.xis  of 
t  ho  curve ;  hence  y  =  U, 


ite  0.    Hcnc*  (1) 


(1) 


cr  limits. 


irfaco  formed  bj 
jasc. 


fi 


lay 


•faoe  formed  by 

vis. 

j  ou  the  axis  of 


KXAMl'hKS. 

The  equation  of  the  generating  curve  is 

X 


a 


Here 


which  in  (1)  gives 


ds  =  \/^a  a;"*  dXf 

•^0 


125 


yx' 

\^yx^  —  ^J\^  dyV 

\\yx^  -  \Jx^j'%a^^x  dx^ 
[2^3:8^  —  2  /  V^rt  —  X dxT 

2TTrt  (2«)»  -  I  (2rt)t 

„       15t  — 8 
=  ^  «   3:r  -4- 

3,  Find  the  centre  of  gravity  of  tlie  surface  formed  by 
the  revolution  of  the  semi-cycloid  round  the  axis  of?/  in  the 
last  example,  \.  c,  round  the  tangent  to  the  curve  at  the 
highest  point. 

Am.   y  =   "  (15t  -  8). 

1  tJ 


126  Al^T   CtrttVKD    SVRPACE. 

84.  Centre  of  G-ravity  of  Any  Curved  Surface.— 

Lot  there  be  a  .shell  hiiving  any  given  cur\ed  -surface  fur 
one  of  its  bounikrie.s;  and  let  /.•  —  the  thickness,  p  —  tlie 
density,  and  ds  n=  the  area  of  an  element  of  the  surface  at 
the  point  [x,  y^  z);  then  (I)  of  Art.  83  becomes 

/  kpx  ds 

^  =  ~. (1) 

/  kp  ds 

and  similar  expressions  for  y  and  a. 

Substituting  the  value  of  ds  (Cal,  Art.  201)  and  cancel- 
ling k  and  p,  we  have 


X  = 


//K'+S  +  S)*^^^''^ 


//(^  +  s  +  ^^)*'^^''^ 


EXAMPLES, 


1.    Find    the  centre  of    gravity  of  one-eighth  of    the 
surface  of  a  sphere. 

Hero  se'  +  f+  z>  z=  a>. 


[    "^  dx"'^  dyV    ~  {a^  -  x^  -  f)*' 


p  r      xdxdy 
J  J  (cvi  —  -a  _  yt)^ 


^mm 


7ed  Surface.— 

ir\cd  surface  for 
ickuoss,  p  =  tlu' 
>f  the  surface  at 
)me8 


(1) 


!01)  and  cancel- 


fa;  dy 
tdy 


B-eighth  of    tho 


i- 


SOLID    OP  liKVOLUTlOS. 


Vil 


First  perform  the  y-intcgra- 
(ion,  :(•  t)oing  constant,  from 
tj  =  0    to   ij  =  U  =  tj^   = 


V«2 


/» ;  tlie  effect  will  be 


10  sum  up  all  the  elements 
>;imilar  to  pq  from  //  to  /. 
L'iie  effect  of  a  subsequent 
./•-i?itegration  will  be  to  sum 
all  these  elemental  strips  that 
are  comprised  in  the  surface  ^. 
of  which  OAB  is  the  projec- 
tion, and  the  limits  of  this  integration  are  a;  =  0  and 
X  =  OA  =  a.    Hence 


S  = 


/^  P»>       xdxdy 

/'a  /'!'"        dxdy   


fj 


nx  dx 


jy 


=  ^ 


dx 


Similarly 


y  =  la,    e  =  irt. 


2.  Find  the  centre  of  gravity  of  one-eighth  of  the  surface 
of  the  sphere  if  the  density  varies  as  the  z-ovdinato  to  any 
point  of  it.     Here  p  =  fiz. 


Ans.  x  =  ^, 


4rt 


2a 


85.  Centre  of  Gravity  of  a  Solid  of  Revolution.— 

Let  a  solid  be  generated  by  the  revolution  of  the  curve,  AB, 
(Fig.  40),  round  the  axis  of  x.  Then  the  elementary 
lectangle,  P^NM,  {— ydx),  generates  an  element  of  the 


^■ta 


H^ 


128 


SOLID   OF  KKVOhVTIOX. 


solid  wliose  volnmo  =  TyS  ijjr  (Cal.,  Art.  203).  Hence  if  the 
density  of  the  solid  is  uniform,  we  have  for  rhe  [)osition  of 
tile  centre  of  gravity  (which  evidently  is  in  the  axis  of  a), 


I  TTtfx  dx         I  y^x  dx 
J;nfdx         J'ifdx 


(1) 


the    integrations    Iwing    extended    over    the  whole  area, 
CABU,  of  the  hounding  curve. 

If  the  density  varies,  the  element  of  muss  may  require  to 
be  taken  ditferentiy.  If  the  density  varies  with  x  alone,  i.  e., 
if  it  is  uniform  all  over  the  r-Mtangular  strip,  PQNAf,  the 
volume  may  be  divided  up  as  already  done,  and  the  element 
of  mass  =  Txpy^  dx.     Hence,  we  shall  have  in  this  caae, 


<r  =: 


/  py^x  dx 
J  Pf  dx 


(2) 


If  the  density  varies  as  y  alone,  we  may  take  a  rectangular 
ebment  of  area  of  the  second  order,  dx  dy,  at  the  point 
{x.  y)  ;  this  area  will  generate  an  element  of  volume 
=  'i-ny  dx  dy  ;  therefore  the  element  of  mass  =  "Z-npy  dx  dy, 
and  wo  have 


J  Jpxydxdy 

J  J pyd^'iy 


(3) 


the  y-integrations  being  performed  first,  from  0  to  y,  the 
ordinate  of  a  point  l\  on  the  bounding  curve ;  and  then 
the  a;-integrationa  from  iiii  to  01). 


I).  Hence  if  the 
■  the  position  of 
the  axis  of  a), 


(1) 


he  whole  area, 

1  may  require  to 
ath  a;  alone,  i.  e., 
ip,  PQNM,  the 
iind  the  element 
n  this  case, 


(2) 


ke  a  rectangular 
ly,  at  the  point 
ent  of  volume 
s  =  'inpy  dx  dy. 


(3) 


rom  0  to  y,  the 
urve ;  and  then 


EXAMPLES. 


EXAMPLES 


129 


1.  Find  tlio  rontio  of  jrnivity  of  the  hoinisphoro  ponerntwl 
1)V  tiu-  ivvoliitioM  of  till'  (luadraut,  AD,  (Fijj.  ;{!)),  nmiid  <)A 
(taken  iis  axis  of  x).  (1)  wlicn  the  dciisitv  is  niiiforin  :  {'l) 
wlieii  it  is  coiislant  over  a  scotion  ))erpiMidiciilar  to  OA  and 
varii's  a-!  tiio  distance  of  this  section  IVoiu  01):  {'■'>)  wlicii 
it  is  constant  at  the  same  distance  from  OA  and  varies  as 
tliis  distance. 


± 


(1)  From  (1)  we  have 


X  = 


I  y^x  dx 
J  y"^  dx 


Putting  X  =.  r  cos  (9,  and  y  =  r  sin  0,   where  r  is  the 
radius  of  the  circle    and  integrating  between  0  =  0  and 

d  =     ,  wo  have 

x  =  |r. 

(2)  Since  p  =  /«,  we  have  from  (2) 

Px^  dx 


X  = 


f 


xy^  dx 

which  gives  «  =  A'*- 

(3). Since  p  =  ny,  wc  have  from  (3) 

r  I'xy^dxdy         fx^dx 


t>  = 


r  I'y^dxdii  I'fdx 


^^ 


••^'^  KXAMPLtJS. 

and  tlie  previous  substitutions  for  x  and  y  give 

a;  =    —  -• 

157T 

2.  Find  tlie  centre  of  gravity  of  a  ])araholoid  of  revolu- 
tion,  tiio  iengtii  of  whose  axis  is  //.  A71S.  x  =  U. 

3.  Find  the  centre  of  gravity  (1)  of  a  portion  of  a  prolate 

splieroid,  the  lengtli  of  whoso  axis  measured  from  the  vertex 

is  c,  and  (:i)  of  a  iienii-spiieroid. 

-        , ,  V  _       c  8rt  —  3c     ,  ,  _ 
Am.   (l).  =  -^-.^^___;(2).^|«. 

86.  Polar  PormulaB.— Let  a  solid  be  generated  by  the 
revolutioi.  of  AB,  (Fig.  41),  round  the  axis  of  a:.  Then  the 
elementary  rectanj:le,  abed,  whoso  mass  =  pr  dO  dr,  (Art. 
HI),  the  thickness  being  omitted,  generates  a  ring  which  is 
an  element  of  the  solid  whose  volume  =  2irr  sin  dpr  dOdr' 
and  the  abscissa  of  the  centre  of  gravity  of  the  ring  is 
r  cos  d.    Hence  (1)  of  Art.  77  becomes 


X  = 


I  I  p)-^  sin  6  cos  d  dd  dr 
J  J  pr'i  sin  e  dd  dr 


(1) 


in  which  p  must  be  a  function  of  r  and  8  in  order  that  the 
integrations  may  be  effected. 

If  the  density  depends  only  on  the  distance  from  a  fixed 
point  in  the  axis  of  revolution,  this  point  may  be  taken  as 
origin,  and  p  will  bo  a  function  of  r  ;  if  the  density  depends 
only  on  the  nistance  from  the  axis  of  revolution,  r>  will 
be  a  function  of  ;•  sin  0. 

EXAMPLE. 

The  vertex  ot  a  right  circular  cone  is  in  the  surface  of  a 
sphere,  the  axis  of  the  cone  coinciding  with  a  dianietirof 


vHyrrtK  of  ouavity  of  am-  souik 


V.W 


rive 


oloid  of  rovolii- 
ns.  i  z=  |//. 

'ion  of  ji  prolate 
from  the  vertex 

(2)  X  =  |a. 

n  era  ted  by  the 
'far.  Then  the 
prdOflr,  (Art. 
I  rintr  which  is 
sin  OprdOdr; 
of  the  ring  ig 


(1) 


Jrder  tliat  the 

'  from  a  fixed 
V  be  taken  as 
■nsity  dej)ends 
hjtion,  .)  will 


3  snrfaee  of  a 
I  diainetc  r  of 


the  sphere,  the  base  of  the  eone  being  a  portion  of  the  sui- 
fiioe  of  the  sphere.  Find  the  distance  of  the  centre  of 
gravity  of  tlic  cone  from  its  vertex,  2a  being  its  vertical 
angle,  and  «.  the  radius  of  tlie  sphere. 

Here  the  /•-limits  are  0  and  in  cos  d  ;  the  fi-hmits  are  0 
and  «;  fj  is  constant;  hence  from  (1)  we  have 


m  = 


I     I    1^  sin  d  cos  0  JO  dr 

•'o_^o 


/•*  sin  d  dd  dr 


=  * 


5=  \a 


f    (2a  cos  ey  sin  0  cos  6  dd 
«{o 

/'"(2rt  cos  0)3  sin  e  dd 

/   cos*  6  sin  6  d9 
''o 

/^cos3  0  sin  6  dd 


1  —  cos'  a 
1  —  cos*  « 


0. 


87.  Centre  of  Gravity  of  any  Solid. — Tjet  {x,  y,  t) 

and  {x  +  dx,  y  +  dy,  z  +  dz)  be  two  consecutive  points  E 
and  F,  (Fig.  42),  within  the  solid  whose  centre  of  gravity  is 
to  be  found.  Tlirough  E,  pass  tliree  planes  parallel  to  the 
co-ordinate  planes  xy,  yz,  zx\  also  through  F  pass  three 
planes  parallel  to  the  first.  The  solid  included  by  these  six 
planes  is  an  infinitesimal  i)arallelopiped,  of  which  E  and  F 
are  two  opposite  angles,  and  the  volume  =  dxdydz.  If  p 
is  the  density  of  the  body  at  E,  the  element  of  mass  at  E 
=z  pdx  dy  dz.  Iloncc  the  co-ordinates  of  the  centre  of 
gravity  of  the  solid  are  given  by  the  equations 


132 


EXAMPLES. 


»  = 


V  = 


f  = 


III  P'-  dx  (ly  dz 
I  I    I  pdx  dy  dz 

J  J  Jpydxdydz 
J  I  I  pdxdy  dz 

I  I  I  pz  dx  dy  dz 


(1) 


(2) 


(3) 


I  I    I  p  dx  dy  dz 
the  integrations  being  extended  over  the  whole  solid. 


EXAMPLES. 

1.  Find  the    entre  of  gravity  of  the  eighth  part  of  an 
ellipsoid  included  between  its  three  principal  planes.* 

Let  tiie  equation  of  the  ellipsoid  be 

t   ,   ?/'   ,   ^'  _  1 
a»  "^  //«  "•■  c»  ~ 

Here  the  limits  of  the  ^-integration  are 

which  call  2,  and  0  ;  the  limits  of  y  are 

Z;  =  3  (l  _  ^)*   and  0, 
which  call  y^  and  0 ;  the  x-limits  are  a  and  0. 


♦  PlaiieH  of  xy,  yz,  tx. 


MM 


POL  AH   ELEMENTS   OF  MASS. 


133 


(1) 


(2) 


(3) 


le  solid. 


ith  part  of  iin 

pIlUU'S.* 


First  integrate  witii  respect  to  z,  and  we  obtain  the 
infinitesimal  prismatic  column  whose  base  is  PQ,  (Fig.  A:'i), 
and  whose  height  is  Pjo.  Then  we  integrate  with  respect 
to  y,  and  obtain  the  sum  of  all  the  columns  which  form 
(lie  elemental  slice  llplmq.  Then  integrating  with  respect 
to  X,  we  obtain  the  sum  of  all  the  slices  included  in  the 
s  did,  OABC.  Hence  (1)  becomes,  since  the  density  is 
uniform, 


If      I    X  dx  dy  dz 

•'z  -'z    ''o 

III    dxdydz 
*^o  «/o    ^0 


-Adx 


Similarly 


y  —  \b,     z  =  \c. 


2.  Find  the  centre  of  gravity  of  the  solid  bounded  by  the 
planes  z  —  (3x,  z  =  yx,  and  the  cylinder  y^  =  'iax  —  a?. 

5« 


Ans.  X  =  ja;  y  =  0;  c 


8 


{d  +  y). 


88.  Polar  Elements  of  Mass.— Let  Fig.  43  repre- 
sent the  portion  of  the  volume  of  a  .soHd  included  between 
its  i)ounding  surface  and  three  rectangular  co-ordinate 
phines. 


T 


134 


POLAR  ELEMENTS  OF  MASS. 

2 


(1)  Through  the  axis  of  z  draw 
&  series  of  consecutive  planes,  divid- 
ing the  solid  into  wedge-shaped 
slices  such  as  COBA. 

(3)  Round  the  axis  of  z  describe 
a  series  of  right  cones  with  their 
vertices'  at  0,  thus  dividing  each 
slice  into  elementary  pyramids  like 
0-PQST. 

(3)  With  0  as  a  centre  describe 
a  series  of    canssecutive  spheres; 
thus  the  solid  is  divided  into  elementary  rectangular  par- 
allelopipeds  similar  to  abpt,  whose  voliune  =  aj)  •  ps  •  st. 

Let  XOA  =  ({>,     COP  =  6,     Op  =  r, 

AOB  =  cl(t>,  POQ  =  fie,  pa  =  dr. 

Then  pq  is  the  arc  of  a  circle  whose  radius  is  ;•,  and  the 

angle  is  dO  ;  therefore 

pq  =  rde. 

Also  ps  is  the  urc  of  a  circle  in  which  the  angle  is  d(t>, 
and  the  radius  is  the  perpendicular  from  ]>  on  OZ,  or 
r  sii  0;  therefore 

ps  =  r  sin  0  d<l>. 

Therefore  the  volume  of  the  elementary  parallclopipcd  = 

»•»  sin  e  dr  dO  d(p ; 

and  if  /)  is  the  density  of  the  solid  at  p,  the  element  of 
mass  is 

pr*  sin  0  dr  dO  d<p. 

Also   the  co-ordinates  of  the  centre  of  gravity  of    this 
element  are 


f  sin  0  cos  <p,    r  sin  0  sin  0,    uud    r  cos  0 ; 


Fig.  43 


ctanj 
ap. 

?ular  pnr- 

ps .  St. 

r, 

dr. 

is  ;•, 

and  tho 

e  angle  is  dtp, 
V  on  OZ,  or 

illelopipcd  = 
3  plement  of 

■ity  of   this 


EXAMPhtlS.  135 

hence  for  the  centre  of  gravity  of  the  whole  solid  we  have 
/    /    /  p/-3  sin2  B  cos  1^  dr  dd  d<t> 
.     J  J  J  Pi"^  '^i»  <^  <''•  f''^  # 

f  /'  /'pf^  sill*  0  sin  </»  rfr  f/0  rf0 


y  = 


fl  = 


J' J'  /'pr^  sin  ddrd0di> 

I  I  I  ^'^  ^'"  ^  ''^^  ^  ''^  ^^  ^'^ 
J'J'fpr'^  sill  '^  <?'•  M  d(f> 


the  limits  of  integration  being  determined  by  the  figure  of 
tiie  Roliil  considered. 

'i'hc  isnglos,  0  and  di,  are  sometimes  called  the  co-latitude, 
and  longihidc,  respectively. 

EXAMPLES. 

1.  Find  tiie  centre  of  gravity  of  a  hemisphere  whose 
density  varies  as  the  nth  power  of  the  distance  from  tho 
centre. 

Take  the  axis  of  z  perpendicular  to  the  jtlane  base  of  tho 
Iu'niis])hore.  Let  n  r=:  the  nitlius  of  the  sphere,  and 
^1  r=  //;■",  where  //  is  I  lie  density  at  tlu'  units  distance  from 
the  <eiilr('.  Firsi  integrate  with  respect  to /•  from  0  to  rr. 
and  we  olitiiin  tiic  iiilinitesiinid  pyramid  O  I'QS'I'.  'I'licn 
inlfgnilc  with  respect  lo  b  from  0  lo  Jt,  and  we  obtain  I  lie 
^•lllll  of  ill!  the  pyramids  which  form  the  elemental  slice, 
COMA,  'riieii  iiilcgratinn  \ntli  icspcct  to  «/>  from  0  to  'It. 
vve  ol)tiiin  the  sum  of  ;dl  (he  slices  included  in  the  hemi- 
sphere.    Ileucej 


136  SPECIAL  METHODS. 

rHn   p\w    /ill 

if  r"^^s\nOcoad(/rd0dfb 

''a    ''o    ''a  '^ 


$  -- 


(*2t     /'It     /la 


I  r"*»  sin  e drdO d<b 

Oq  i/q  i/q  ^ 

„      /      /     sin  0  cos  9  fZe  ^0 

pin      /ijrr  > 

/      /     sill  0  dd  d<l> 


n  4- 
— ; — id' 


?j  +  3    a 
.*.    «  = .  -• 

??  +  4    2 ' 

and  it  is  clear  that        i  =  y  ~  0. 

2.  Find  the  oontre  of  gnivify  of  a  portion  of  a  solid 
sphere  contained  in  a  rijrlit  cono  wlioso  voricx  is  the  centit 
of  tiie  sphere,  the  density  of  the  solid  varying  as  the  wtli 
power  of  tlie  distance  from  tlie  centre,  the  vertical  angle  of 
the  cone  being  =  'iic,  iiud  tlie  radius  =  a. 

Take  tho  iixis  of  I  lie  coni^  us  tluit  of  z,  iiud  tiuy  pkuo  through  it  as 
tliat  from  which  hmgitiule  Is  ineiiBiircd. 

Am.  z  =  ^^-^2  ^^  +  ^''^  ")'  '^'^'^  x  =  yz=0. 

89.  CIpecial  Methods.— In  tiie  preceding  Articles  we 
Inive  given  the  usual  forniuia^  for  linding  the  centres  of 
gravity  of  bodies,  but  particular  cases  may  occur  which  may 
be  most  conveniently  treated  by  special  methods. 


EXAMI^LES. 

1.  A  circle  revolves  round  a  tangent  line  through  an 
angle  of  180°;  lind  the  centre  of  gravity  of  the  solid 
generated. 


■«^ 


r  (IB  d<t> 


Hon  of  a  solid 
'X  is  (lie  centi-c 
'ing  ii«  the  wtli 
ertic'iU  angle  of 

ano  through  it  as 


ng  Articles  wo 
the  -^eiitix's  of 
cur  which  may 
ods. 


10  through   an 
v  of   tlio  nolid 


JiXAMPLES. 

Let  OY  be  the  tangent  line  about 
which  the  circle  revolves,  and  let  the 
plane  of  the  paper  bisect  the  solid ;  the 
centre  of  gravity  will  therefore  lie  in 
the  axis  of  x.  Let  P  and  Q  bo  two 
consecutive  points  ;   and  let  OM  =  x, 

and   MP  =  »/  =  V^n^-^^.      The 


137 


Fi8.44 


elementary  rectangle,  PQqp,  will  gen- 
erate a  semi-cylindrical   shell,  ^vhose  volume  =  ^yirxdx, 
the  centre  of  gravity  of  which  will  be  in  the  axis  of  a;  at  a 

distance  —  from  0  (Art  78,  Ex.  1,  Cor,).     Heuce, 


TT 


/      —  Zy  Ttx  ax 
/     2y  nx  dx 

r:r^  '^/'iax  —  x^  dx 
-    „  5a 


/    X  '^"iax  —  :< 


t^  dx 


2rr 


3.  Find  the  centre  of  gravity  of  a  right  pyramid  of  uni- 
lorm  density,  whose  base  is  any  regular  jilane  figure. 

Let  the  vertex  of  the  pyramid  l)e  the  origin,  and  the  axis 
of  the  pyramid  the  axis  of  a;;  divide  the  })yramid  into  slices 
of  the  thickness  dx  by  planes  perpendicular  to  the  axis. 
IMien  us  the  areas  of  these  sections  arc  as  the  squares  of 
their  honuilogous  sides,  and  as  the  sides  are  as  their  dis- 
tances fiom  tlie  vertex,  i^o  will  the  areas  of  the  sections  be  as 
thvi  sciuares  of  their  distances  fnmi  the  vertex,  and  therefore 
the  masses  of  the  slices  are  as  the  S(|uares  of  their  distances 
tVoni  liu' vertex.  Now  inuigine  each  slice  to  be  condensed 
into  its  centre  of  gravity,  which  point  is  on  the  axis  of  ,'. 
Then  the  problotn  is  reduced  to  tindiug  the  centre  of  grav 


^tM 


138 


THEOREMS   OF  PAPPOS. 


ity  of  a  material  line  in  which  the  density  varies  as  the 
square  of  the  distance  from  one  end,  and  which  may  be 
found  as  in  Ex.  6,  (Art.  78).  (Jailing  a  the  altitude  of  the 
pyramid,  we  have 


y?  dx 


y" 


—  =  |a. 


o^dx 


which  i."?  the  same  as  in  Art.  75. 

90.  Theorems  of  Pappus.*— (1)  If  a  plana  curve 
revolve  round  any  axis  in  its  plane,  the  area  of  the 
surface  generated  is  equal  to  the  length  of  the 
revolving  curve  multiplied  by  the  length  of  the 
path  described  by  its  centre  of  gravity. 

Let  s  denote  the  length  of  the  cnrve,  x,  y,  the  co-ordinates 
of  one  of  its  points,  ^,  y,  the  co-ordinates  of.  the  centre  of 
gravity  of  the  curve;  then,  if  the  curve  is  of  constant 
thickness  and  density,  we  have  from  {i)  of  Art.  78, 


y  = 


I  yds 

Jds 


2nys  =  %-tT  I  y  ds\ 


(1) 


the  second  member  of  which  is  the  area  of  the  surface 
generated  by  the  revolution  of  tlio  curve  whose  length  is  .v 
about  the  axis  of  x,  (Gal.,  Art.  11)3)  ;  and  the  first  nioniber 
is  the  length  of  the  revolving  curve,  s,  inulU{)lio(l  l)y  the 
length  of  the  path  described  by  its  centre  of  gravity,  v'rry. 


♦  Usually  calloci  (Juldln'H  Thporoms,  but  orlKli.ally  uimnclalixl  by  I'appiiH.    (800 
Waltou'i)  Mechanical  ProblouiB,  p.  it,  8d  Ed.) 


r  varies  as  the 
which  may  be 
iltitude  of  the 


plants  curve 
5  area  of  the 
n,0h  of  the 
ngth  of  the 


le  co-ordinates 
the  centre  of 
is  of  constant 
rt.  78, 


(1) 

if  tlio  surface 
<c  length  is  H 
first  nioniher 
ipliod  1)}'  the 
ravity,  'In]/. 

t\  by  Pappus.    (8oo 


TUEOKEMS   Of  I'Al'l'US. 


139 


y  = 


or 


(2)  //'  a  plane  area  revolve  round  any  axis  in  its 
plane,  the  volume  generated  is  equal  to  the  area  of 
the  revolving  figure  maltiplied  by  the  length  of  the 
path  described  by  its  centre  of  gravity. 

Let  A  denote  tlie  phme  area,  and  let  it  bo  of  constant 
thickness  and  density,  tlien  (2^  of  Art.  83  becoinea 

y ^y"?/  (Ix  dy 

I  I  dx  dy 

2Try  C  Cdk  =  2rtJ  J  ydx  dy, 

(substituting  dk  for  dx  dy), 

.-.     ■■ZnyA  =  TT  hfdx,  (3) 

the  integral  being  taken  for  every  point  in  the  perimeter  of 
tlic  urea;  but  the  second  member  is  the  volume  of  the 
solid  generated  by  tlie  revolution  of  the  area  (Cal.,  Art. 
20;3) ;  and  the  first  member  is  the  area  of  tlie  revolving 
figure,  A,  multiplied  by  th_e  length  of  the  path  described 
ijy  its  centre  of  gravity,  2TTy. 

Cou.— If  the  c\irve  or  area  revolve  through  any  angle,  0, 
instead  of  2tt,  (1)  and  (2)  become 


6 


and 


=  dj'y  ds, 
df/A  =  ^ej'y^dx, 


(3) 
(4) 


and  the  theorems  are  still  true. 

Sen.— If  thi'  axis  cuts  the  revolving  curve  or  area,  the 
tiu'orems  still  apply  will)  the  couvenlion  tiiat  the  surface 
or  volume  generated  by  the  portions  of  the  curve  or  area  on 
opposite  sides  of  the  axis  are  alleeted  with  opposite  signs. 


mm 


i  I 


uo 


EXAMPLES. 


I   i 


ll 


\    I 


. 


<    I 


I    t 


EXAMPLES. 

1.  A  circle  of  radius,  a,  revolves  round  an  axis  in  its  own 
plane  at  a  distance,  c,  from  its  centre;  lind  tiie  surface  of 
the  ring  generated  by  it. 

The  leng'Ji  (circuniferonfc)  of  liie  revolving  curve  — 
2rra;  the  length  of  clie  patii  described  by  its  centre  of 
gravity  =  2tc  ; 

.  • .     the  area  of  the  surface  of  the  ring  =  ^tt^oc. 

2.  An  ellipse  revolves  round  an  axis  in  its  own  piano, 
the  perpendicular  distance  of  wliicli  from  the  centre  is  c\ 
Jiiid  the  volume  of  tlie  ring  generated  during  a  complete 
revoLiiion. 

Let  a  and  b  be  the  semi-axes  of  the  ellipse;  then  tiu> 
revolving  area  =  wi ;  the  length  of  the  path  described  by 
its  centre  of  gravity  =  'iic  ; 

•  •.    the  volume  of  the  ring  =  2rr^aic. 

Observe  tliat  tUo  voliimo  is  the  same  for  any  position  of  the  nxes 
of  tlie  ellipse  with  respect  to  tlie  axis  of  revolution,  provided  the  per 
pendicular  distance  from  tlmt  axis  to  tlie  centre  of  the  ellipse  is  the 
same. 

3.  The  surface  of  a  sphere,  of  radius  a,  =  4Tr«2;  the 
lengih  of  a  semi-cirouniferenre  =  ttk  ;  lind  the  length  of 
the  ordinate  to  the  centre  of  gravity  of  the  arc  of  a  semi- 


circle. 


.I«v.  7/  = 


4.  The  volume  of  a  sphere,  of  radius  a,  —  ^m^ ;  the 
ar  'a  of  a  semicirel'  =  j^mi^:  lind  tiie  .  stance  of  the  centre 
of  gravity  of  the  semicircle  from  ^ae  diametei, 

A  us.  y 


ill 


5.   A  circular  lower,  thc^  diameter  of  whici;  is  20  li..  i.s 
being  bu  it,  and  for  every  fool  it  .'ses  it  inclines  1  in.  from 


axis  in  itis  own 
tlie  surface  of 

Iving  curve  — 
y  its  centre  of 

its  own  plane, 
ho  centre  is  c  ; 
ing  ii  complete 

ipse;  then  tlio 
b  described  by 


sition  of  tho  nxes 
irovided  tho  per 
the  ellipse  is  the 

,  =  4Tr«2;    the 

the   length  of 

lire  of  u  semi- 

'wv.   ;/  =  

--=    fTTft^;      the 

p  of  the  centre 


ia 


ns.  y  =  .,   - 

Cili  is  5J(i  I'l. .  is 
nes  1  in.  from 


KX.\MP!jES. 


141 


the  vertical;  find  the  greatest  height  it  can  reach  ,vitnout 
falling.  ^•'"■^■-  "^-iOft. 

(i.  A  circular  tal)le  weighs  20  lbs.  and  rests  on  /our  legs 
in  its  circumference  forming  a  square  ;  find  the  least  ver- 
tical pressure  that  must  be  apjtlied  at  its  edge  to  overturn  it. 

Am.  20(\/:i  +  1)  =  48.28  11)8. 

7.  If  the  sides  of  a  triangle  be  3.  4.  and  5  feet,  find  the 
distance  of  the  centre  of  gravity  from  each  side. 

.4,v«.  |,  1,  i  ft. 

8.  7\n  equilateral  triangle  stands  vertically  on  a  rough 
plane  ;  tind  the  rati:)  of  the  heiglit  to  the  base  of  the  plane 
when  the  triangle  is  on  the  point  of  overturning. 

Ans.  •v/3  :  1. 

0.  A  heavy  bar  14  feet  long  is  bent  into  a  right  angle  so 
that  the  lengths  of  the  portions  which  meet  ut  the  angle 
are  8  feet  and  t!  feet  respectively;  show  that  the  distance 
of  the  centre  of  gravity  of  the  bar  so  bent  from  the  point 
of  the  bar  which  was  the  centre  of  gravity  when  the  bar 

9  V2 


was  straight,  is 


feet. 


10.  An  equilateral  triangle  rests  on  a  sqnare,  and  the  base 
of  the  triangle  is  equal  to  a  side  of  the  squf.re  ;  find  the 
centre  of  gravity  of  the  figuie  thus  formed. 

Ans.  At  a  distance  from  the  base  of  the  triniiglc  equal  to 

3 

___ of  the  base. 

8  -f   3  \/3 

11.  Find  the  inclination  of  a  rough  plane  on  which  half 
a  regular  hexagon  can  just  rest  in  a  vertical  position  with- 
out overturning,  with  the  shorter  of  its  parallel  sides  in 
contact  with  tlie  ])lane.  Auk.  W  \/'5  :  ■^^ 

12.  A  cylinder,  the  diameter  of  which  is  10  ft.,  and  height 
60  ft.,  rests  on  another  cylinder  the  diameter  of  which  i:: 


m 


^ 


142 


EXAMPLES. 


1<S  ft.,  atid  height  G  ft.;  and  their  axes  coiiicido  ;  find  their 
coininMn  centre  of  gravity.   Aus.  -iTy?  ft.  from  the  ba.se. 

1:5.  Into  a  Jiullow  cylindrical  vessel  11  ins.  hiirh,  and 
weighing  lU  ll).s.,  the  centre  of  gravity  of  whicli  is  T)  ins. 
from  tile  base,  a  iiniforin  solid  cylinder  (J  ins.  long  and 
weighing  -^O  lbs.,  is  just  fitted;  find  their  common  centre  of 
gravity.  Anf<.  ;j§  ins.  from  base. 

14.  The  middle  points  of  two  adjacent  sides  of  a  siiiiare 
arc  joined  and  the  triangle  formed  by  this  straight  line  and 
the  edges  is  cut  off;  lind  the  centre  of  gravity  of  the 
remainder  of  the  s(jiuire. 

.Ins.   y,\  of  diagonal  from  centre. 

15.  X  trapezoid,  whose  ])arallel  sides  are  4  and  l:i  ft. 
Jong,  and  the  other  sides  eacii  equal  to  5  ft.,  is  i)lace<l  with 
its  plane  vertical,  and  with  its  shortest  side  on  an  inclined 
plane  ;  find  the  relation  between  the  height  and  base  <,f  the 
jilaiie  wiien  the  trapezoid  is  on  the  point  of  falling  over. 

A  us.  8  :  7. 

I'i.  A  regular  hexagonal  prism  is  i)laced  on  an  inclined 
l)lan<>  with  its  end  faces  vertical  :  find  the  inclination  of 
the  plane  so  that  tiie  prism  may  just  tumble  down  the  plane. 

Ans.  30". 

K.  A  regnlar  polygon  just  tumbles  down  an  inclined 
plane  whose  inclination  is  10°  ;  how  many  sides  has  tlie 
l'»'ygon  ?  A>u^.   18. 

18.  From  a  sphere  of  radius  /»'  is  removed  a  sphere  of 
radius  /•,  tlie  distance  between  their  centres  being  c  ;  find 
the  eeiitre  of  gravity  of  tlie  remainder. 

.ins.   h  is  1)11  (he  line  joining  their  centres,  and  at  a  dis- 


tance 


rr 


/.•^' 


from  tlu'  centre. 


l'».    A    rod   of   uniform   thickness   is    made   up  of   cipial 
■ngtiis  of  three  snlistances,  the  densities  of  which  taken  in 


i(l'> ;  find  their 
um  tlie  base. 

ins.   liigli,   iinJ 

liicli   in   ')  ins. 

ins.    Ion;,'  iind 

unon  c'cnti'fof 

j.  from  base. 

JS  of  ii  8<iiiart' 
■aight  line  and 
gravity  of  the 

from  cuntro. 

4  and    1:J  ft. 

is  i)lacc'<l  with 
n  an   inclined 
nd  l>aso  cf  the 
illing  over. 
Ans.  8  :  7. 

on  an  inclined 

inclination  of 

own  the  plane. 

Ans.  30". 

n  an   inclined 

sides  has  tlio 

A /If.    18. 

:!d  a  sj)here  of 
being  c ;  fintl 

and  at  a  dis- 


n\^  of   ei|nal 
hieh  taken  in 


EXAMPLES. 


143 


order  arc  in  the  proportion  of  i,  2,  and  :}  ;  fmd  the  position 
of  th.e  centre  of  gravity  of  the  rod. 

Am.  At  I'Jg  of  the  whole  length  from  the  end  of  the 
densest  part. 

20.  A  heavy  triangle  is  to  be  suspendctl  by  a  string  iiass- 
ing  through  a  point  on  one  side  ;  determine  the  position  of 
tiic  point  so  t'lat  the  triangle  may  rest  with  one  side 
vertical. 

Ans.  The  distance  of  the  point  from  one  end  of  the  side 
.—  twice  its  distance  from  the  other  end. 

21.  The  sides  of  a  iieavy  triangle  are  3,  4,  .').  respectively  ; 
if  it  l)e  susiiended  from  the  centre  of  the  inscrilH'<l  circle 
show  that  it  will  rest  with  the  shortest  side  horizontal. 

22.  The  altitude  of  a  right  cone  is  //,  and  a  diameter  of 
the  base  is  h ;  a  string  is  fastened  to  the  vertex  and  to  a 
l)()int  on  the  circumference  of  the  circular  base,  and  is  then 
put  over  a  smooth  peg;  show  that  if  the  cone  rests  with  its 
axis  horizontal  the  length  of  the  string  is  ^{li^  +  &). 

23.  Find  the  centre  of  gravity  of  the  helix  whose  equa- 
tions are 

a;  =  rt  cos  <^  ;     y  =  «  sir^  0  5     ^  =  ht'P. 

y     -  a  —  X    .  __  z 

Ans.  .«  =  ka-^  ;  y  =  A-'"  T^""'  "  '~  i' 

34  Find  the  distance  of  the  centre  of  gravity  of  the 
c.atenarv  (Cal.,  Art.  1T7),  from  the  axis  of  /.the  curve 
being  divided  into  two  e(Hial  portions  l)y  the  axis  ot  //. 

ins.   If  -11  is   the   length  of  the  curve  and   (A, /•)  is  the 
extn>mity.  tlie  centre   of  gravity  is  on   the  axis  of  ^  at  a 

distauce  ^^-J^  -  iV..m  the  axis  of  x. 


l-l*  KXAMI'LES. 

'i').  Find    the   centre   ol"  f,'nivity   of  tlie  urea   iucludeil 

between  the  are  of  the  ])ariib(.Ia,  //-'  =  hix,  and  tlie  strai^rhi 

line  y  =  kx.  .        .        8«      .        -iu 

Ans.  .  =  ^,  y  = -^. 

:e(;.   Find   tlie  centre  of  gravity  of  the  urea  bonnded  bv 
•h"  cissoid  and   it^i  asymptote,  the  equation  of  the  cissoiil 

beiujj  ifi  =^ .  J,,..    ;  J,, 

"Z".  Find  tiio  centre  of  gravity  of  liie  area  of  the  witcli 
of  Agnes  i. 

Anx.  At  a  distance  from  tiie  asymptote  equal  to  \  of  the 
diameter  of  the  biuse  circle. 

;.'S.  Find  the  centre  of  gravity  of  the  area  inchided  be- 
tween tlie  arc  of  a  semi-cycloid,  the  circumference  of  the 
generating  circle,  and  the  base  of  the  cycloid,  the  common 
tangent  to  the  circle  and  cycloid  at  the  vertex  of  the  hitter 
lieing  taken  as  axis  of  x,  the  vertex  being  origin,  and  a  the 
radius  of  the  generating  circle. 


'o 


29.  Find  the  centre  <.f  gravity  of  tlie  area  contained  be- 
tween the  curves  if  =:  a.c  and  f  -  2ax  —  x-,  which  is 
above  the  axis  of  x.     ,        .  I.jt  _  44    .  (,, 


A  IIS.    i   :=   I! 


1-,T^40'  ^  -  ;}Tr 


30.  Fiml  the  centre  of  gravity  of  the  area  included  by 
the  curves  f  =  a:   and  x^  —  hi/. 

Alls.  :•  .—  2V<ai:';  y  =  -^^ahK 

;n.  Find  the  distance  of  the  centre  of  gravity  of  the  area 
of  the  circular  sector,  BOCA,  (Fig.  3!)),  from  the  centre. 
Let  2d  —  the  angle  included  by  the  bounding  radii. 

I        -        „    «in  0 
Ans.  X  —  |rt  - n~- 


XAilPLES. 


145 


10   area   iuchnlod 
,  and  the  strai^dil 

^^¥'y  =  -lc' 

area  boiiiulcd  hy 
on  of  tlio  cirsoid 

A  us.  X  =z  |r/, 
iroa  of  tho  witcli 
fiiual  to  \  of  tlic 

rca  included  be- 
[imference  of  the 
L)id,  the  com  moil 
tex  of  the  latter 
)iigin,  and  a  the 

-«        -        « 

—  '■<;  y  =  pt- 

ea  contained  be- 
;  —  X'-,  which  is 
a 


y 


rca  included  by 

;  y  =  A«'*^- 

ivity  of  the  area 
mi  tlie  centre, 
idiii":  radii. 

-    sill  0 


32.  Find  Llie  distance  of  the  centre  of  gravity  of  the 
circular  segment,  BCA,  (Fig.  39),  from  the  centre. 

a  i'in^  0 
Ans.  X  =  \ 


B(f 


y  _  sin  (7  cos  0       1'^  area  of  ABC 


33.  Find  the  centre  of  gravity  of  the  area  bounded  by 
the  cardioid  r  =  a{\  +  cos  b).  Ans.  x  =  ^a. 

34.  Find  the  centre  of  gravity  of  the  area  included  by  a 


loop  of  the  curve  r  —  a  cos  20. 


Ans.  X  = 


128flj\/2 


35.  Find  the  centre  of  gravity  of  the  area  included  by  a 


loop  of  the  curve  r  =  a  cos  39 


81a  \/3 
Ans.x  =  —^-. 


36.  Find  the  centre  of  gravity  of  the  area  of  the 
sector  in  Ex.  31,  if  the  density  varies  directly  as  the  dis- 
tance from  the  centre.                          .,„    .  _  3rt  sin  d 

4  d 

37.  Find  the  centre  of  gravity  of  the  area  of  a  circular 
sector  in  which  the  density  varies  as  the  wth  power  of  the 
distance  from  the  centre. 

Ans    ^^-  •  ^,  where  a  is  the  radius  of  the  circle,  I  tho 
n  +  3      I 

length  of  the  arc,  and  c  the  length  of  the  chord,  of  the 
sector. 

38.  Find  the  centry  of  gravity  of  the  area  of  a  circle  in 
which  the  density  at  any  point  varies  as  the  nlh  power  of 
the  distance  from  a  given  point  on  the  eircunifereiice. 

Ans.   It  is  on  the  diameter  passing  through   the  given 

2  (n  +  2) 
point  at  a  distance  from  this  point  c(pial  to  — — r^"  "'' 

a  being  the  radius. 
7 


itaii 


14«; 


EXAMPLKS. 


3!).  Find  tlie  centre  of  gmvity  uf  the  area  of  u  ([uadrant 
of  an  ollii)se  in  which  tiie  density  at  any  point  varies  as 
the  distance  of  the  point  from  tlie  major  axis. 


A)US.   X  :=z  ^11  ;   y  =z 


Sir 


40.  Find  the  distance  of  tlio  centre  of  gravity  of  the  sur- 
face of  a  cone  from  the  vertex. 


Let  a  r=  the  altitude. 


Ans.  X  =  la. 


41.  Find  Uu  centre  of  gravity  of  the  surface  formed  by 


revolving  the  curve 


r  =z  a{\  +  CCS  6), 


round  the  initial  line. 


A71S.    X  = 


50a 


i'i.  A  parabola  revolves  round  its  axis;  find  the  centre 
of  gravity  of  a  jtortion  of  the  surface  between  the  vortex 
and  a  plane  i)erj)tndicular  to  the  axis  at  a  distance  from 
the  vertex  oipial  to  f  of  tiie  latus  rectum. 

A/IS.  Its  distance  from  the  vertex  —  ^}j  (latus  rectum). 

43.  Find  tlie  centre  of  gravity  of  a  cone,  tlie  density  of 
each  circular  slice  of  which  varies  as  the  nth  power  of  its 
distance  from  a  parallel  jilane  through  the  vertex. 

Let  the  vertex  be  the  origin  and  a  the  altitude. 

Ans.  X  :=  — —  a. 
M  +  4 

44.  Find  the  centre  of  gravity  of  n  cone,  the  density  of 
every  particle  of  whicii  increases  as  its  distance  from  the 
axis. 

Anx.  :7  — -  I'U  where  I',.,-  vertex  is  the  origin  and  a  the 
al'iiliuK'. 

4').  Find  the  cenln'  ol'gra\ity  of  the  volume  of  uniform 
density  contained  between  ii  lieniis|ihere  and  a  cone  whose 
vertex  is  the  vertex  of  the  honiispliere  and  base  is  the  base 
of  the  hemisphere. 


1  of  ii  (iuadrant 

'  point  varies  as 

is. 

,.       _       3::  , 

l'<-^y=  j^  b. 

ivity  of  the  siir- 


ins.  X  =  l-rt. 
face  formed  by 


}IS.    X 


50a 


find  the  centre 
iveeii  the  vortex 
a  distance  from 

liitus  rectum). 

tlie  density  of 
h  power  of  its 
L'rtex. 
tude. 

n  +  3 

the  density  of 
tance  from  tiio 

f^'in  find  (I  the 

inc  of  Mniloi'Mi 
I  11  cone  uhdsc 
)aso  is  tlio  base 


EXAMPLES. 


147 


Ans.  X  —     ,  where  the  vertex  is  the  origin  and  a  tiic 
altitude. 

40.   Find  the  distance  of  tlie  centre  of  gravity  of  a  lienii- 
sphere  from  tlie  centre,  the  radius  being  a. 

Aus.    i  =.  |«. 

47.  Find  tlie  centre  of  gravity  of  the  solid  generated  by 
the  revolution  of  the  semieycloid, 

y  ■=.  V^fta;  —  j^  ■\-  a  vers"'  - , 

(1)  round  the  axis  of  x,  and  (2)  round  the  axis  of  y. 

An..  (1)  X  =  i^--^---  |A^  ;    (.>)  ,  =:  \^-  +  ~J-^-. 

48.  Find  the  centre  of  gravity  of  the  volume  formed  by 
the  revolution  round  the  axis  of  a:  of  the  area  of  the  curve 


3/<  —  axy"^  +  vS  -z  0. 


3fflTr 
Ans.  X  —   .•--• 


49.  Find  the  centre  of  gravity  of  the  volume  generated 
by  the  revohition  of  the  area  in  Ex.  2"J  round  the  axis  of  y. 

ba 


Ans.  V  ~  -. 


50.  Find  the  centre  of  gravity  of  a  hemisjihere  when 
the  density  varies  as  the  s<iuare  of  the  distance  from  the 
centre.  ,,„,    -  _  •'". 


A  ni 


Ans.  X 


'A.  Find  the  centre  of  gravity  of  ll\c  solid  gcncnilcd  l)y  a 
semi-|iarabola  bounded  by  the  latus  rectum,  ivvolving 
round  the  latus  rectum. 

Ans.  Distance  from  focus  -  /j  of  latus  rectum. 


148  EXAMPLES. 

52.  Tho  voriifs  of  x  right  circular  cone  is  at  the  centre  of 
M  spliyrc ;  And  the  centre  of  gnivity  of  a  body  of  iiniforni 
ilcns^ity  oontained  witiiiu  the  ( oiu'  and  the  spliere. 

Aus.  The  disianco  (if  [he  oi'iitre  of  gravity  from  tlie  ver- 
tex of  tho  cone  =  -.,-  (1  -h  cos  a),  where  «  =  the  st-ini- 

Cl 

tertical    angle   of  the   cone   and   a  —.  the   radius  of  the 
sphere. 

515.  Fiiul  the  distance  from  tho  origin  to  the  centre  of 
gravity  of  the  solid  generated  by  the  revolution  of  the 
cardioid  round  its  prime  radius,  its  equation  being 

r  ==«(!  +  cos  0). 

A  US.  X  =  Ja. 

54.  Find  by  Art.  00  (1)  the  surface  and  (^Z)  the  volume 
of  the  solid  formed  l)y  tho  revolution  of  a  cycloid  round 
the  tangent  at  its  vertex. 

Alts.  Surface  =  ^'^d^;  Volume  =  rrV. 

55.  Find  (1)  the  surface  and  (;i)  the  volume  of  the  solid 
formed  by  the  revolution  of  a  cycloid  n)und  its  base. 

Ann.   (1)  \<Ti,fi\  (^>)5tV'. 

5G.  An  e(iuilateral  triangle  revolves  round  its  base, 
whose  length  is  (t  ;  find  (1)  tho  area  of  the  surface, 
and  (•^)  the  volume  of  the  llgun!  desci-ihed. 

Ans.  (!)  ira-i  \^'6  ;  (2)  '"'^. 

57.  Find  (1)  the  surfiu-e  and  {'i)  the  volume  of  a  rinu 
with  a  circular  section  wln)se  inlermd  diameter  is  I'j  ins., 
and  Ihickness  ;{  ins. 

Alls.   (1)  444.1  s.|.  in.;  (2)  333.1  cub.  iu. 


s  at  the  centre  of 
,  body  of  uniform 
fipliero. 
ity  from  tlic  ver- 

c  fc  =  tlio  st-nii- 

10   radius  of  the 

to  tlie  centre  of 
evolution  of  the 
>n  being 

A  US.  X  =z  ^a. 

1  (3)  tlie  volume 
ii  cycloid  rouiul 

olume  =  Tr2«3, 

lume  of  the  solid 
d  its  base. 

round  its  base, 
of  the  surface, 
3d, 

^  V3  ;  (2)  ^. 

volume  of  a  riii'i 
amcter  is  12  ins., 

333.1  cub.  in. 


CHAPTER    V. 

FRICTION. 

91.  Friction. — Friction  is  thai  force  which  acts  betweer 
two  bodies  at  their  surface  of  contact,  and  in  the  direction 
of  a  tangent  to  tl\at  surface,  so  as  to  resist  their  sliding-  on 
each  other.  It  depends  on  the  force  with  which  the  bodies 
are  pressed  together.  All  the  curves  and  surfaces  which  we 
have  hitlu  ito  considered  were  supposed  to  be  smooth,  ami, 
as  such,  to  offer  no  resistance  to  the  motion  of  a  body  in 
contact  with  them  in  any  other  than  a  normal  direction. 
Such  curves  and  surfaces,  however,  are  not  to  be  found  in 
nature.  Every  surface  is  cai)al)le  of  destroying  a  certain 
amount  of  force  in  its  tangent  plane,  i.e..  it  i)ossesses  a  certain 
degree  of  roiajlincxs,  in  virtue  of  wiiich  it  resists  the  sliding 
of  other  surfaces  upon  it.  Tiiis  resistance  is  cidled ///'//ow, 
and  is  of  two  kinds,  viz.,  slidimj  and  rolliwj  friction.  The 
first  is  that  of  a  heavy  body  dragged  on  a  plane  or  other 
surface,  an  axle  turning  in  a  fixed  box,  or  a  vertical  shaft 
turning  on  a  horizontal  plate.  Friction  of  the  second  kind 
is  that  of  a  wheel  rolling  along  a  plane.  Hoth  kinds  oi 
friction  are  governed  by  the  same  liiws;  tiie  former  is  much 
greater  than  the  latter  under  the  same  circumstances,  and 
is  the  only  (me  that  we  shidl  consiilcr. 

A  Kinnoth  surface  is  one  which  opposes  no  resistance  to 
the  motion  of  a  body  np.'ii  it.  A  roxfili  surface  is  oiu' 
which  does  oppo.se  a  resistaiu'C  to  the  mntion  of  a  body 
ujton  it. 

riic  Hiirl'iircs  of  all  li()ili;'s  ("insist  of  very  Htnall  «>levritions  iiml 
(it'pi-t'csion.H,    HI)    tliiit,    if   tliin-  arc  prcssi'!    npiiiint    cacli    oilier,    tlui 

oh'vatioiiH  of  om-  fit.  mor •  Ii'sh.  into  tlip  (I'Mivc-si'ms  <f  tin"  i  tlicr, 

luid  till' Hiirfiici'ii  iiit('rii:'iiriiii;u  .'Ui'li  iitliur;  aiid  tlic  uuiluul  peiu'tru- 


150  LAWS   OF  FlilCTIOS. 

tioii  is  of  courso  grnntcr,  if  tlic  iiressiiig  forco  is  greater.  Hence, 
when  a  force  is  apiilietl  so  as  to  cause  one  body  to  move  on  anotlier 
with  wliicli  it  is  in  eoiitact,  it  is  necessary,  liefon;  motion  can  take 
|)lace,  eillier  to  break  otttlie  elevations  or  coiniiress  tlieni,  or  force  tlie 
iMxlies  to  se|iarate  far  enough  to  aUow  tiieiii  to  pass  imic'i  otlier. 
Much  III  tliis  ;w'y//;'('.'<.i  niiiy  be  removed  by  polishing;  and  the  etl'ect 
of  niucli  of  it  may  bi;  destroyed  by  lultriciition. 

Friction  always  acts  along  a  tangent  to  the  surface  at  the  |)oiut  of 
contact  ;  and  its  direciion  is  opposite  to  tliat  of  tlu'  line  of  motion  ;  it 
presents  itself  in  the  motion  of  a  body  aa  a  passive  force  or  resistance," 
since  it  can  only  hiiidi  r  motion,  but  can  never  produce  or  aid  it.  In 
investigations  in  mechanics  it  can  be  considered  as  a  force  acting  in 
opposition  to  every  motion  whose  direction  lies  in  the  plane  of  contact 
of  the  two  bodies.  Whatever  may  bt*  the  direction  in  which  we  move 
a  body  resting  upon  a  horizontal  or  inclined  plane,  the  friction  wi'll 
always  act  in  the  opjjosite  direction  to  that  of  the  motion,  i.  e.,  when 
we  slide  a  boily  di.vii  an  inclined  plane,  it  will  a])pcar  ns  a  force  up 
the  plane.  A  surface  may  also  resist  sliding  motion  by  means  of  thj 
adhesian  between  its  substance  and  that  of  another  body  in  contact 
with  it.f 

Tbo  friction  of  a  body  on  a  surface  i.s  measured  by  the 
least  force  wiiicli  will  put  the  body  in  motion  along  tiic 
surfiice. 

92.  Laws  of  Friction. — In  our  ignorance  of  the 
constil lit  1011  of  biidie.'^.  the  laws  of  friction  must  be  deduced 
froiii  experiment.  Experiments  made  by  Coulomb  and 
Morin  have  established  the  following  laws  of  friction : 

(1)  The  fridioii.  varies  as  the  normal  pn^si^ure  when  the 
matcriah  of  ttw  surf'acfs  in  con  fact  rrmain  the  same.  Sub.se- 
ijiient  experiineutii  have,  liowever,  considerably  modified 
this  law,  and  shown  thiii,  it  can  be  regarded  only  as  tin 
ai)pr()xiniation  to  the  truth.  Wh>n  the  'iressure  is  very 
great  it  is  found  that  the  friction  is  less  than  this  law 
would  give. 

*  WcUbiich,  p.  .son. 

t  St'c  liaiikiiicH  .\|)|>IU!(I  MeclinnicD,  p.  800. 


I  greater.  Hen«\ 
o  niovt^  on  miotlKT 
i(  luotior.  can  take 
tliciii,  or  t'orct'  llii' 

pass   t'lic'i    otlicr. 
iig,  and  the  t'lli'ct 

face  lit  the  |)oint  of 
line  of  motion  ;  it 
orce  or  resistance,* 
dace  or  aid  it.  In 
IS  a  force  acting  in 
lie  plane  of  contact 
in  which  we  move 
II',  the  friction  wi"]! 
motion,  i.  e.,  when 
)pear  as  a  force  up 

II  by  means  of  thj 
lur  body  in  contact 


ncasurofl  by  the 
otioii  along  tlic 


lorance  of  tho 
mist  bo  dcdiKvil 
•  Coulomb  and 
f  friction : 

n^si^ure  when  the 
'ic  .wwr,  Siib.sc- 
prably  modified 
•dod  only  as  an 
tressure  is  very 
I  tlian    this  law 


iw. 


LAWS   OF  FKIVTIOX. 


151 


(2)  77/1™  frirfioii-  is  indppcndcnl  of  tJic  extent  of  the  sur- 
fdirx  ill  loiilfirt  so  loiifj  as  tJic  iioriiiaf  iirrssinr  iritiaiiis  the 
smiic.  When  the  stiri'accs  in  contact  are  very  small,  as  ibr 
'!isl;iiice  a  cylinder  resting  on  a  surface,  tiiis  hiw  gives  tlie 
''riction  mucli  too  great. 

'I'liesc  two  laws  are  tnie  when  the  body  Is  on  the  point  of  moving, 
and  also  when  it  is  actually  in  motion  ;  but  in  the  case  of  motion  the 
magnitude  of  the  friction  is  not  always  the  same  as  when  the  body  is 
'jei/iiuiitif/  to  nu)Ve  ;  when  tliere  is  a  difference,  the  friction  is  greater 
in  the  state  bordering  on  motion  than  in  a'"  lal  motion. 

(3)  The  friction  is  independent  of  the  velocity  when  the 
l/ody  is  in  motion. 

It  follo\v.s  fnmi  these  laws  that,  if  R  be  tlie  normal 
pressure  l)etween  the  bodies,  /'  the  force  of  friction,  and  /i 
the  constant  ratio  of  tlie  latter  to  tho  former  when  slipping 
is  about  to  ensue,  we  have 


F  =  (lU. 


(1) 


The  fraction  /t  is  called  the  cr  fficieiit  of  friction  ;  and  if 
the  first  law  were  true,  //  wouhl  1)0  strictly  constant  for  the 
same  pair  of  Ijodies,  wliatever  the  magnitude  of  the  normal 
])ressure  between  them  migiit  be.  This,  iiowever,  is  not 
tlie  case,  Wlien  tlie  normal  jiressure  is  nearly  e(|Uid  to  that 
wliicli  would  crush  cither  of  the  siirlaces  in  conttict,  tho 
force  of  friction  increases  more  rapidly  than  the  normal 
pressure.  E(|tiation  (1)  is  nevertheless  very  nearly  true 
when  the  difFerences  of  normal  pressure  are  not  very  great  ; 
and  in  what  follows  wo  shall  assume  this  to  I)e  the  case. 

Hem  AUK. — The  laws  of  friction  were  eatal)lished  by  Coulomb,  a 
distinguished  Fn'uch  officer  of  Engineers,  and  were  founded  on 
exi)eriments  made  by  him  at  Hix-hefort.  The  n'sulta  of  these  oxpori 
ments  were  presented  in  I7S1  to  the  French  Academy  of  Sciences,  and 
in  ITB")  his  Memoir  on  Friction  was  publi^■lled  A  very  full  abstract 
of  this  jmper  is  given  in  I>('  Young's  Nutnrnl  fldlnKophji,  Vol.  II, 
ji  170  (Ist  Rd.),  Further  exiieiiments  were  nuide  at  ,Metz  by  Moiin, 
1831-34,  by  direction  of  the  French  military  authorities,  ihi^  rertult  of 


153 


ANOLK   OF  FlfrCTlGN. 


wliich  has  hocn  to  confirm,  with  slight  fxcoptions,  nil  the  results  o< 
Couloml).  iind  to  (Ictcriniiic  witli  coiisidcrnMc  prcrisinn  the  numerical 
vhIui's  of  tiic  coctlicii-nts  o!'  friction,  for  all  the  siibstnnccs  usiiiillv 
employed  iti  the  coiislniction  of  machines.  (See  (iaibraith's  Me 
clianics,  p.  (is,  TwisJeii's  Practical  iMechaiiics,  p.  l;i8,  and  WeisbuchV 
M'cliaiiicri,  \'ol.  I,  ]).  ;il7.) 

93.  Magnitudes  of  Coefficients  of  Friction.— I 'rac 

ticiillv  tluTc  is  no  observed  cofllicie'iil  mucli  greator  lliaii  1. 
Most  of  the  onliiuiry  coetHc'ioiits  are  loss  tlian  ^.  Tlie  fol- 
lowing msulfs,  selected  from  a  table  of  coefficients,*  will 
iitford  an  idea  of  the  amount  of  friction  as  determined  by 
experiment  ;  tlieso  results  apply  to  the  friction  of  motion. 

For  iron  on  stone      /t  varies  between  .i?  and  .7. 
For  timber  on  timber "      "  "        .2  and  .5. 

For  liml)er  on  metals  "      "  "        .3  and  .6. 

For  metals  on  metals  "      "  "      .15  and  .25. 

For  full  ])articidars  on  this  subject  tlio  student  is  referred 
to  Itiiiikine's  Applied  .Mechanics,  p.  209,  and  Moseley's 
Engineering,  p.  124,  also  to  the  treaise  of  M.  Morin,  where 
he  will  find  the  subject  investigated  in  all  its  completeness. 

94.  Angle  of  Friction.— 77/ r  aiir/If  at  which  a  rov(ih 
pliow  or  surface  iiinij  he  inclined  xo  ihnl  a  hndy,  when  acted 
upon  by  the  force  of  (jravily  only,  may  just  re,"!  upon  it  with- 
out aliding,  ih  called  the  Aityle  of  Friction.  \ 

Let  «  bo  the  angle  of  inclination  of 
the  i»lane  AH  just  as  the  weight  is  on 
the  point  of  slipping  down ;  IT  the 
weight  of  the  body  ;  I!  the  iu)rmal  pres- 
sure on  the  plane;  F  the  force  of  fric- 
tion acting  along  the  jila-.te  =  fiJ?  (Art. 
f)2).  Then,  resolving  the  forces  along  and  porpondicular  to 
the  plane  we  have  for  e(iiiilil)rium 


Fi9.45 


•  Raiikliic'x  Applied  Mochiiiilcs,  p.  811. 

t  SDint'ilinos  calk'il  "  llio  aiiijle  of  rt'pose;"  iil^o  cnllcd  "  tli«  "'ailtlng  aiifilc  nf 

TCBielHIlCC." 


iiMi 


ns,  nil  tlio  results  o{ 
■i'-inii  the  nuiiicriciil 
siibslnticcs  usiiiilly 
vr   (iiilbniith's    ,\l( 
loH,  and  WeisliacliV 


nic- 


Friction.— I 

•li  <>iViilor  than  1. 
tlmii  i.  The  Jbl- 
c'oertioii'iifs,*  will 
as  (loteriniiic'd  by 
;tion  of  motion. 

n  ..5  and  .7. 
.2  and  .5. 
.2  and  .6. 
.15  and  .2ri. 

tiidont  is  roforrt'd 
!),  and  Mosok-y's 
'  M.  Morin,  whore 
i(.s  coniph'tcnoss. 

if  which  n  rou(/h 
'  hi)d]i,  whpu  arled 
red  upon  it  with- 
t 


porpondicnhir  to 


'  llu!  ''.iiltliiK  aiiKlc  of 


REACTION  OF  A    KOVOH  CCRVE. 


uR  =:  U'sin  «  ;  A'  =:  IT  cos  «; 


tan  a  ==  /t, 


153 


(1) 


which  jrivcs  tlio  limiting  value  of  the  inclination  of  the 
plane  fur  which  e(|uilil)riuin  is  possible.  The  body  will  rest, 
on  the  plane  when  the  angle  of  inclination  is  less  than  the 
angle  of  friction,  and  will  slide  if  the  angle  of  inclination 
exceeds  that  angle;  and  this  will  be  the  case  however  great 
W  may  be  ;  the  reason  being  that  in  whatever  manner 
we  increase  \V,  in  the  same  i)r()portion  we  increase  the 
friction  upon  the  plane,  which  servos  to  prevent  M'from 
sliding. 

From  (1)  we  see  that  the  tangent  of  the  angle  of  friction 
js  equal  to  the  coefficient  of  friction. 

95.  Reaction  of  a  Rough  Curve 
or  Surface. — Let  Ali  be  a  rough  curve 
or  surface ;  P  the  position  of  a  particle 
on  it ;  and  siii)pose  the  forces  acting  on 
P  to  be  conlined  to  the  plane  of  the 
paper.  Let  A*i  =  the  normal  resistance  of  the  surface, 
acting  in  the  nornuil,  PN,  and  F  —  the  force  of  friction, 
acting  along  the  tangent,  Pl\ 

'J'he  resultant  of  7^*1  and  Z^,  called  the  Total  Resistance* 
of  the  surface,  is  represented  in  magnitude  and  direction  by 
the  line  J'R  =  R,  which  is  the  diagonal  of  the  parallelo- 
gram determined  by  7?i  and  F.  We  have  seen  that  the 
total  resistance  of  a  smooth  surface  is  normal  (Art.  41)  ;  but 
this  limitation  does  not  a})ply  to  a  roiitjh  surface.  Let  0 
denote  the  angle  between  R  and  the  normal  R^  ;  then  <^  is 
given  by  the  equation 


tan  0  = 


R, 


*  Htncliln'H  Simlcs,  p.  54. 


154 


FRtCTlON  0.\  Ay  r.CLlNEV   1  LANK. 


Ileiico,  (f)  will  bo  a  maxiniuri  when  the  force  of  fr;  'on, 
/',  hojirs  the  greatest  ratio  to  (lie  normal  nressiire  li,.  IJiit 
this  greatest  ratio  is  :!ttai',>.(l  whe:.  tie  lu-ily  is  just  on  the 
point  of  slipp'ig  along  Hie  surface,  and  is  what  we  called 
the  eoetlicient  oFfrietion  (Art.  Wi),  thr.t  is 


=  ^; 


.'•    tan  (j)  z=  fi. 

Therefore  the  greatest  nwjh'  by  which  the  Total  Resistance 
of  a  rough  curve  or  surface  can  denate  fro7n  the  normal  is 
the  angle  tvhose  tangent  is  the  roefficient  of  friction  for  the 
bodies  in  contact  j  and  this  deviation  is  attained  when  slip- 
ping is  about  to  coinmence. 

Cor.— By  (1)  of  Art.  94,  tan  «  =  /*; 

.'.     ^  =  ce; 

hence,  the  direction  of  the  total  resistance,  R,  is  inclined  at 
an  ai'iglf  a  co  the  normal  ;  i.e.,  the  greatest  angle  that  the 
Total  Resistance  of  a  rough  curve  or  surface  can  make  with 
the  normal  is  equal  to  the  angle  of  friction,  corresponding 
to  the  two  bodies  in  contact. 


96.  Friction  on  an  Inclined  Plane. — A  body  rests  on 
a  rough  inclined  plane,  and  is  acted  on  by  a  given  force.  /', 
in  a  vertical  plane  which  is  perpendicular  to  the  inclined 
plane;  find  the  limits  of  the  force,  and  the  angle  at  which 
the  least  force  capable  of  drawing  the  particle  up  the  plane 
must  act. 

Let  i  =  the  inclination  of  the  plane  to  the  horizon  ;  Q  = 
the  angle  between  the  inclined  plane  and  the  line  of  action 
of  P\  n  =  the  coeilicieiit  of  friction  ;  and  let  ns  first  sup- 
pose that  the  body   is  on   the  jxiint  of  nioviuf^  down  the 


'.ANK. 


FiiicriOy  <>i\  .  -  ixcjj.YKD  ri-AM-:. 


ir,', 


force  of  fri  'on, 
ressiire /(',.  Hut 
ly  is  just  ou  tlio 
s  what  we  called 


Total  Resistance 
m  the  normal  is 
friction  for  the 
lined  when  slip- 


U,  is  inclined  at 
/  amjle  that  the 
■e  can  make  with 
n,  corresponding 

A  body  rests  on 

I  given  force,  /', 

to  the  inclined 

angle  at  wliich 

ilc  up  the  plane 

le  horizon  ;  6  = 
10  line  of  action 
let  ns  fu'st  snp- 
oviii'T  down  the 


plane,  so  that  frict'  >n  ir  a  force  acting  n\}  the  plane,  tiicii 
resolving  along,  and  perpendicular  to,  the  plane,  we  have 

/r'  ^  /'  ^.^^fi  0  —  11  ■  sin  t, 

li  +  r  sin  0  =  W  cos  i, 

F  =  nE; 

sin  /  —  /t  cos  / 


.-.    P  =W 


cos  6  —  n  sin  0 


And  if  P  is  increased  so  that  motion  up  the  ])1ane  ."'i., 
begintiing,  F  acts  in  an  opposite  direction,  and  there.  ^ 
the  sign  of  ju  must  be  changed  and  we  have 


P  =  W 


sin  i  +  /'  cos  i 
cos^  +  /i  sin  S 


(2) 


Hence,  there  will  be  equilibrium  if  the  body  be  acted  on  by 
a  force,  the  magnitude  of  which  lies  between  the  values  of 
P  in  (1)  and  (2).  Substituting  tan  <p  for  fi  (Art.  95) ;  (2) 
becomes 

P^W"^^^-  (3) 

cos  (0  —  o) 

To  determine  6  in  (2)  so  that  /'  shall  l)e  a  minimum  we 
must  i)ut  the  first  derivative  of  /'  with  respect  to  0  =  0, 
therefore 


dP  rrr  /   •       •    .  .,       Siu  »  -  /*  COS  6 

de  =  ^  (8in  i  +  ficos  t)  ^^^^  f+T^^-e]^ 

.'.    tan  0  =  /t ; 


0; 


that  is,  the  force  P  noces-ary  to  draw  the  body  up  the  plane 
will  be  the  least  possit)lo  w'ii.'n  0  =  the  angle  of  friction. 


^^ 


15G 


DOUBLK-iycI.ISi:!)   PLAXE. 


He'-ce  wc  infer  tliat  ii  given  force  act.*  to  tlie  greatest 
adviintiigo  in  driiggiug  a  weiglit  up  a  hill,  if  tiie  angle  .it 
wliicli  its  line  of  action  is  inclined  to  tiic  liili  is  e([iiai  to 
the  angle  of  friction  of  the  liill.  Similarly,  a  force  acts  to 
the  greatest  advanlage  in  dragging  a  weiglit  along  a  hori- 
zontal plane  if  lis  line  of  action  is  inclined  to  the  plane 
•It  the  angle  of  frittioiiof  the  plane.  We  may  al-so  (h'ti-r- 
mine  from  this  the  angle  at  which  the  traces  of  a  drawing 
horse  should  he  inclined  tw  the  plane  of  traction. 

These  results  are  those  which  are  to  he  ex})ected,  because 
some  part  (jf  the  force  ought  to  lie  expended  in  lifting  the 
weight  from  the  plane,  .so  that  friction  may  be  diminishetl. 
(Sec  Price's  Anal.  Mech's.  Vol.  I,  p.  100.) 


97.   Friction  on    a   Double-Inclined  Plane.— Two 

bodies,  whose  weights  are  /'  and  Q.  rest  on  a  rough  double- 
inclined  [ilane,  and  are  connected  by  a  string  which  passes 
over  a  smuolh  jK'g  at  a  point,  A,  vertically  .over  the  intersec- 
tion. U.  of  the  two  planes,  Find  the  position  of  equili- 
brium. 

Let  f<  and  ji  be  the  inclinations  of 
the  two  planes  ;  let  I  —  the  length  of 
the  string,  and  //  —  Al};  and  let  6 
and  0  be  the  iUigles  vhe  porlior.s  of 
the  string  make  with  the  planes. 

Suppose  P  is  on  tiie  point  of 
ascending,  and  (>  of  (k.<reti(lui(j. 
Then,  since  the  motion  of  each  body  is  about  to  er.sue,  the 
total  resistances.  R  and  -S  must  each  make  the  angle  of 
friction  with  the  corresponding  normal  (Art.  95,  Cor.) ;  and 
since  the  weight.  /*,  is  about  to  move  u|)wards  the  friction 
must  act  downwards,  and  therefore  J>  must  lie  below  the 
normal,  while,  since  Q  is  about  to  move  downwards,  the 
friction  mnst  act  vijjwards,  and  therefore  ^S'  must  be  above 
the  nnrmal. 


Fig.47 


to  tlio  greatest 

if  tlu'  aii<i;le  at 

lull  i.s  eijUiil  t'l 

I  foire  acts  to 

il   ailing'  a   linri- 

umI   to  the  piano 

may  also  d'/tcr- 

cc'S  of  a  drawing 

ction. 

xpocted,  hecaiKC 
ed  in  lit'tinjj  the 
\-  be  diniinished. 


a  Plane.— Two 

a  rougii  doiiblo- 
ng  wliieli  i)a.sses 
over  tlie  intersec- 
isitioii  of  equili- 


mt  to  or.siio,  the 
ike  tlio  angle  of 
■t.  95,  Cor.);  and 
ards  the  friction 
ust  lie  below  the 
downwards,  the 
)'  must  be  above 


D()rnLK-L\( ■i.i.\i:d  i'laxk. 


157 


If  T  is  the  tension  of  the  string,  we  have  for  the  equi- 
librium of  i',  (Art.  ;J2), 

~       cos  (0  -  </)) 
.Vnd  for  the  equilibrium  of  Q, 

^  ^  cos  (B'  +  <t>) 
Equating  the  values  of  7'  we  get 

,sin  («  +  0)  _  ^^-^iii  (l^  —  '/») 


cos  {()  —  <p)         '  cos  {()'  +  <\>y 


0) 


and  if  /'  is  alioiit  to  niovi'  duini  tiir  plane  the  friction  acts 
in  an  oi)posite  direction,  and  ther-fore  the  sign  of  0  must 
be  chunged  and  we  have 

sin  (a  -9)  ^  ^sin  (/^  +  0)  _  /g) 

cos  (ft  +  (!>}        ^cos  {0'  -0) 

(1)  or  (2)  is  the  only  statical  e(|uation  connecting  the 
given  (luantitics. 

We  obtain  a  geometric  CKjuation  by  expressing  the  length 
of  the  string  in  terms  of  //,  «,  li,  d,  and  «',  which  is 


,  /c(»s  (c    ,    cos  li\ 
Vsiu  0        sill  0  / 


(3) 


From  (1)  or  (•.')  and  (:i)  the  values  of  0  and  0'  can  be  found, 
and  this  determines  the  positions  of  /'  and  Q. 

Ol/icrifisf  111  us  : 

InsU-ad  of  cftnsidering  the  total  resistances,  /.'  and   N.  we 
mav  consider  two  normal  resist :.n(!es.  //,  and  .S',,  and  two 


158 


DOVRI.K-I.XCIjISKn    I'hA  M:. 


forces  of  friction,  |lR^  and  //>',,  acting  respectively  down 
tlie  plane  «  and  up  the  plane  [i.  In  this  case,  considerin-,' 
the  equilibriiini  of  /'.  and  resolving  forces  along,  and  per- 
pendicular to,  the  plane  a,  we  have 


n  «  +  /«/?,  =  Tcose,  ) 
IS  «  =  III  +  2' sin  e,    \ 


(4) 


and  for  the  etiuilibrium  of  Q, 


<2sin/3  =  iiS^  +  TcosO',  ) 
^coS)3  -  <S',  +  T&mO'.    \ 


(5) 


Elimiiuiting  ^,,  .v,.  and  T  from  (4)  and  (r))  we  get  (1), 
the  same  statical  equation  as  before. 

The  method  of  considering  total  resistnnrcs  instead  of 
their  normal  and  tangential  components  is  usnally  more 
simple  than  the  separate  consideration  of  the  latter  forces. 
(See  Minchin's  Statics,  p.  GO.) 

Cor.— If  Q  is  given  and  P  be  so  small  that  it  is  about  to 
ascend,  its  value,  I\,  will  be  given  by  (1), 


^        ^  sin  («  +  <p)  cos  (6'  +  tpy 


(6) 


and  if  P  is  so  large  that  it  is  about  to  drag  Q  up,  its  value, 
Pgj  will  be  given  by  (2) 


Q 


sin  (fi  -\-  (/))  cos  (0  -f-  </>) 
sin  («  —  0)  cos  (O'  —  0) 


(7) 


the  angles  0  and  0'  being  connecled  by  (.'{). 

'i'hei'c  will  be  e(|uilibri!ini  if  f,H)c  acted  on   by  any  force 
whose  magnitude  lies  between  /',  and  /%. 


'spectively  down 
ase,  considering' 
along,  and  per- 


(4) 


;        (5) 

(5)  wo  get  (1), 

inri's  instead  of 
is  nsnally  more 
!)e  latter  forces. 


at  it  is  about  to 


0) 
1>) 

(6) 

<2iip, 

its 

value, 

1>) 

(7) 

1   by 

any 

force 

FRICTION   OF  A    Th'i'iVyioy. 


159 


Fig.48 


98.    Friction  on  Two  Inclined    Planes.— A  beam 

rests  on  two  rough  inclined  planes;  find  the  position   of 
ei|uilil)rium. 

Let  a  and  b  be  the  segments,  AG 
and  BG,  of  the  beam ;  let  6  be  the 
inclination  of  the  beam  to  the  hori- 
zon, u  and  i3  the  inclinations  of  the 
planes,  and  R  and  -S'  the  total  resist- 
ances. Suppose  that  A  is  on  the 
j)oint  of  ascending;  then  the  total 
resistances,    R    and    S,   must    each 

make  the  angle  of  friction  with  the  corresponding  normal 
and  act  to  the  right  of  the  normal. 

The  three  forces,  H',  R,  S,  must  meet  in  a  ;  nt  0  (Art. 
G'^) ;  and  the  angles  GOA  and  GOB  arc  diual  to  «  +  <t>, 
and  (i  -  <p,  respectively. 

Hence     {a  +  h)  cot  BGO  =  a  cot  GOA  —  5  cot  GOB, 
or      {a  +  I)  tan  0  =  a  cot  («  +  <t>)  —  i  cot  (/i  -  <p).     (1) 

CoK.— If  the  planes  ai-e  smooth,  0  =  0,  and  (1)  becomes 

(«  -f-  i)  tan  0  =  rt  cot  a  —  J  cot  &. 

(See  Ex.  7,  Art.  62.) 

99.  Friction  of  a  Trunnion.*— T/w/jiM/ows  are  the 
cijUndncal  projcclions  from  the  cmh  o/afshnff,  which  rest 
on  the  rourarc  xiirfncex  of  cijUndrical  boxes.  A  shaft  rests 
in  M  horizontal  position,  with  its  trunnions  on  rough 
cyh.  irieal  surfaces;  find  the  resistance  due  to  friction 
whieh  is  to  lie  overcome  when  the  shaft  begins  to  turn 
about  a  horizontal  axis. 

*  gometimcH  called  "  Journal." 


•^    • 


IGO 


FlilCTIOX  OF  A    PIVOT. 


Fig.49 


Let  JA(7and  BAED  bo  two  rijxlit 
sections  of  the  trunnion  and  its  box; 
tiie  two  cireies  are  iiuigeut  to  each 
other  internally.  If  n<»  rotation 
takes  place  the  trunnion  presses 
upon  its  lowest  iioint,  II,  through 
which  the  direction  of  the  resulting 
pressure,  li,  passes ;  if  tlie  shaft 
begins  to  rotate  in  the  direction  vVlI,  the  trunnion  ascends 
along  the  inclined  surface,  EAB,  in  conse((uencc  of  the 
friction  on  its  bearing,  until  the  force,  S,  tending  to  move 
it  down  just  balances  the  friction.  /'.  Resolving  R  into  a 
normal  force  A' and  a  tangential  ojie,  (V,  we  have,  since  the 
;angential  conijtoncnt^  of  R  \n  urging  the  truuuion  down 
the  surface  =  the  friction  which  opposes  it. 

S  -  F  =  //A';    but     7.'2  =  S'  +  A^'j 

or  R^  =  ifiN-^  +  N^;      ■ 

R 


therefore 

and  the  friction 


N  = 


V'l+ii'' 


F  = 


liR 


R  tan  (f> 


Vl  4-  p'        Vl-f-  tan2</) 


-.-zrr.  (Art.  95), 


or 


/''=  /?  sin  0. 


Hence,  fo  find  Ihc  frirlioii  it/iDii  a  /nnnn'nii,  iniilfipjj/  the 
ri'siilhtnt  iif  f/ic  /(irccs  w/iirh  (tcl  ujmn.  it  by  llie  sine  of  tlm 
inKjtc  (if frirlioH. 


100.    Friction  cf  a  Pivot.— A    hea,\y  circular  shaft 
rests  in  a  vertical  position,  with  its  end,  which  is  a  circular 


Fig.49 


rnnnion  ascMids 
5i'(nioti(jc  of  tho 
tending  to  move 
Giving  R  into  a 
)  hiivo,  since  tlie 
L!  truuuiou  down 


+  ^V»j 


rt.  95), 


'nil,  multiply  flip, 
hij  llie  sine  of  //ui 


y   (.'irculiir   sliaft 
1  it'll  is  a  circular 


FRTCTIO.y   OF  A    PIVOT. 


161 


section,  on  a  horizontal  plate;  find  the  resistance  due  to 
friction  wliich  is  to  he  overcome,  when  tlie  shaft  begins  to 
revolve  about  a  vertical  axis. 

Let  a  be  the  radius  of  the  circular  section  of  the  shaft ; 
let  the  plane  of  (r,  0)  be  the  horizontal  one  of  contact 
between  tlic  end  of  the  shaft  and  the  plate;  and  let  the 
renire  of  tlie  circular  area  of  contact  be  the  pole.  Let 
W  —  the  weipiit  of  the  .«liaft,  then  the  vertical  jiressure  on 

each  unit  of  surface  is  — ii  and  thercioru,  if  r  dr  dO  is  the 

area-clement,  we  h  ive 

W 

the  pressure  op  the  element  ==:     -,,  r  dt  dQ  ; 

,'.     the  friction  of  the  clement  —  /t  --  r  dr  dO. 

The  friction  is  opposed  to  motion,  and  the  direction  of  its 
acti'in  is  taii<>viit  to  the  circle  dcscrilied  by  tlic  clement ; 
the  moment  of  tlie  friction  about  the  vertival  a.xis  through 
the  centre 

_  fiWr'hlrdO 

therefore  the  moment  of  friction  of  tlu'  whole  circular  end 
_     /'^T    ,><i,,  Wr-drdO  _  '^J^IIVf  ,,^ 


and  conse(pieiitly  varies  as  the  radius,  ilence  arises  tho 
advantage  of  I'educiiig  to  the  smallest  jiossible  dimensions 
tlie  area  of  the  base  of  a  vertical  shaft  revolving  with  its 
end  resting  en  a  liorizoiitai  lied. 

From  (1)  we  may  regard  tiie  whole  friction  due  to  the 
pressure  as  acting  at  a  single  point,  and  at  a  distance  <":•.. m 
the  centre  of  motion  ofiual  to  two-thirds  of  tho  raums  of 


1G3 

the  base  of  the  shaft, 
lover  of  friction. 


EXAMPLES. 


This  distance  is  called  the  mean 


When  the  shaft  is  vertical,  and  rests  upon  its  circular  end 
in  a  cylindrical  socket  the  cylindrical  projection  is  called  a 
Pivot. 

EXAMPLES. 

1.  A  mass  whose  weight  is  750  lbs.  rests  on  a  horizontal 
plane,  and  is  pulled  by  a  force,  /',  whose  direction  makes 
an  angle  of  ].5°  with  the  horizon  ;  determine  P  and  the 
total  resistance,  R,  the  coefficient  of  friction  l)eing  .02. 

Ans.  P  :=:  413-3  lbs.;  R  =  75(5.9  lbs. 

3.  Determine  P  in  the  last  example  if  its  direction  is 
horizontal.  Ans.  P  =  405  lbs. 

3.  Find  the  force  along  the  i)lane  recpiired  to  draw  a 
weight  of  25  tons  up  a  rough  inclined  plane,  the  coefficient 
of  friction  being  i^,  and  the  inclination  of  the  plane  being 
such  that  7  tons  acting  along  the  plane  would  support  the 
weight  if  the  plane  were  smooth. 

Ans,  Any  force  greater  than  17  tons. 

4.  Find  the  force  in  tlie  preceding  example,  supposing 
it  to  act  at  the  most  advantageous  inclination  to  the  jdane. 

Ans.   IS-jSj  tons. 

5.  A  ladder  inclined  at  an  angle  of  00°  to  the  horizon 
rest^  between  a  rovf/fi  ]Hi,vement  and  the  smooth  wall  of  a 
house.  Show  that  if  the  ladder  begin  to  slide  when  a  tniin 
has  ascended  so  that  his  centre  of  gravity  is  half  way  uj), 
then  the  coefficient  of  friction  between  the  foot  of  the 
ladder  and  the  j)avement  is  J  VS. 

(I.  A  body  wlioso  weight  if  20  lbs.  is  just  sustained  on  a 
I'nugli  incilined  plane  liy  a  horizontal  force  of  2  ll»s.,  and  a 
force  of  10  11)8.  .ilong  the  plane  ;  the  coefficient  of  friction  is 
'i ;  find  the  inclination  of  the  plane.      Ans.  2  tair'  (J  J). 


ialled  the  mean 

ifs  circular  end 
Hon  is  called  a 


on  a  horizontal 
[lirectiou  makes 
nine  /'  and  the 
1  l)eing  .02. 
=  750-9  lbs. 

its  direction  is 
P  =  405  Iba. 

lirod    to  draw  a 

L',  the  coefficient 

the  piano  being 

uld  support  the 

tlum  17  tons. 

inpie,  supposing 
;;n  to  the  plane. 
/,'>•.    I5^\  tuns. 

'  to  the  horizon 
amoiitlt  wall  oi"  ii 
ide  when  u  tuiin 
(•  is  half  way  u]), 
he  foot  of   the 


t  Hu  stained  on  u 
of  2  ll»s. ,  and  a 
ent  of  IVielioii  is 


EXAMPLES. 


103 


7.  A  heavy  body  is  placed  on  a  rough  plane  whose 
inclination  to  the  horizon  is  sin  '  (|),  and  is  connected  i)y 
a  siring  passing  over  a  smooth  pvilley  with  a  body  of  equal 
weight,  which  hangs  ti-ecly.  Supposing  that  motion  is  on 
the  point  of  ensuing  up  the  plane,  find  the  inclination  of 
tlie  string  to  the  plane,  the  coefficient  of  friction  being  |. 

Ans.  0  =  2  tan-i  (J). 

8.  A  heavy  body,  a(!ted  upon  by  a  force  equal  in  magni- 
tude to  its  weight,  is  just  about  to  ascend  a  rough  inclined 
plane  under  the  influence  of  this  force;  find  the  inclination, 
0,  of  the  force  to  the  inclined  i)lane. 


Ans.  0  z= 


/,  or  20  +  i  —    ,  where  i 

4t 


inclination 


of  the  plane,  and  </>  =:  angle  of  friction.     (9  is  here  sup- 
jmsed  to  lie  measured  from  the  iipiwr  side  of  tl;e  inclined 

Itlane).     If  [^  >  2^  -f  /,  0  is  negative  and  the  ai)plicd  force 

vill  act  towards  the  nnder  side. 

!^».  In  the  first  solution  of  the  last  example,  what  is  the 
magnitude  of  the  pressure  on  the  plane  ? 

Ans.  Zero.     Explain  this. 

10.  If  ilie  shaft,  (Art.  100),  is  a  square  ]>rism  of  the 
weight  W,  iind  rotates  about  an  axis  in  its  centre,  prove 
that  the  moment  of  tlie  friction  of  the  square  end  varies  as 
the  side  of  the  scpiare. 

11,  If  the  shaft  is  composed  of  Iwo  equal  circular 
cylimliM'S  pliiced  side  l)y  side,  and  rotates  about  the  line  of 
contact  of  the  two  cylinders,  show  that  the  juoment  of  the 
fri(!ttoii  of  the  surface  in  contact  with  the  horizontal  plane 

Vi.  What  is  the  least  cocflicient  of  friction  lluit  will 
allow  of  a  heavy  body's  being  just  ke|)t  from  sliding  down 


164  h'XAM/'l.KS. 

ail  inclined  plam'  uf  ^nveii  inclination,   the   body    (whoso 
weight  is  IT)  being  sustained  Liy  a  given  horizontal  force,  /'  ? 

'"..u  i  -P 


Ana 


\\~-[-'  P  tun  % 


13.  It  is  observed  that  a  body  whose  weight  is  known  to 
be  U'can  be  just  sustained  on  a  rough  inclined  plane  by  a 
horizontal  force  P,  and  that  it  can  also  be  just  sustained  on 
the  same  plane  by  a  force  Q  np  the  plane;  exjiress  the 
angle  of  friction  in  terms  of  t  hese  known  forces. 

PW^ 


Aps,  Angle  of  friction  —  cos~i 


14.  It  is  observed  that  a  force,  Q^,  acting  up  a  rough 
inclined  plane  will  just  sustain  on  it  a  body  of  weight  W, 
and  that  u. force,  (>g,  acting  up  the  plane  will  just  drag  the 
same  body  up;  iind  the  angle  of  friction. 

Ans,  Angle  of  friction  :=  sin"'  ^  --^ZL^— 


'ft' 


ViP-^i^, 


15.  A  heavy  uniform  rod  rests  with  its  extremities  on 
(he  interior  of  a  rough  vertical  circle;  (ind  the  limiting 
position  of  e<}uilibrium. 

Ans.  If  '■Ik  is  the  angle  subtended  at  the  centre  by  the 
rod,  and  A,  the  angle  of  friction,  the  limiting  inclination  of 
the  rod  to  the  horizon  is  given  by  the  equation 


,       „  sin  2A, 

tan  0  — 


cos  !i/L  -f-  cos  2tt 

K).  A  solid  triangular  prism  is  placed,  with  its  axis 
horizontal,  on  a  rough  inclined  plane,  the  indinalion  of 
whidi  is  graxluidly  incri'aHc(|  ;  determine  the  natiii'c  of  llic 
initial  motion  of  the  prism. 

Ans.  If  (he  triangle  AIi(!  i^  the  section  perpendicular  lo 
the  axis,  and  the  side  AB  is  in  contact  .vith  the  plane,  A 


ic    body    (whoso 
rizo/ital  force,  Z^? 

\\  +  I^in'i' 

'iylit  is  known  to 
iiied  plane  hy  a 
just  sustained  on 
me;  express  tlie 
'orces. 

PW^ 

tinfjf  up  a  rough 
ody  of  weight  W, 
i'ill  just  drag  the 

ts  extremities  on 
ind  the   limiting 

the  centre  by  the 
iig  inclination  of 
tiou 


KXAMl'LES. 


105 


being  the  lower  vertex,  the  initial  motion  will  be  one  of 
ti'.aibliiig  if 

J2  ^  3(J  _  di 


f^> 


4d 


the  sides  of  the  triangle  being  n,  h,  c,  and  its  area  A.  If  fi 
is  less  than  this  value,  the  initial  motion  will  be  one  of 
slipj.ing. 

IT.  A  frustum  of  a  solid  right  cone  is  i)l;iced  with  its 
base  on  a  rough  inclined  plane,  the  inclination  of  which 
is  gradually  increased;  determine!  the  initure  of  the  initial 
motion  of  the  body. 

yl«,s'.  If  the  radii  of  the  larger  and  smaller  sections  ar<'  R 
and  /•,  and  h  is  the  height  of  the  frustum,  the  initial  motion 
will  be  one  of  tumbling  or  slipping  according  as 

4/.'    jr^  +  nr_+  r"^ 
''^^   h"  m  +  'iRi-  +  3r2" 

IS.  An  elliptic  cylinder  rests  in  limiting  '"quilibrium 
between  a  rough  vertical  and  an  equally  rough  l)ori'.;ontal 
plane,  the  axis  of  the  cylinder  l)eing  horizontal,  and  the 
major  axis  of  the  ellipse  inclined  to  the  liorixou  at  ai»  angle 
of  45°.     Find  the  coefficient  ()f  friction. 

-,  e  being  the  eccentricity 


Alls,  fi  =  — 
of  the  ellipse. 


I.  with  its  axis 
V  inclination  o)' 
le  naiurc  of  llir 

["-'rpendiciilai-  to 
til  the  plane,  A 


lAlMta 


CHAPTER    VI. 


THE    PRINCII-LE    OF    VIRTUAL    VELOCITIES* 

101.  Virtual  Velocity.— If  the  point  of  applicafion  of 
a  force  be  conceived  as  disphiced  'throv(/h  an  indefinitely  .small 
space,  the  resol red  part  of  tlie  displacement  in  the  direction 
of  tlie  force,  is  called  the  Virtual  Telocity  of  the  force ;  and 
tlie  product  of  the  force  into  the  virtual  velocity  has  been 
called  the  virtual  moment\  of  the  force. 

Thus,  let  O  bo  the  original,  and  A 
the  new  ])oint  of  application  of  the 
force,  P,  acting  in  the  direction  OP, 
and  let  AN  bo  drawn  i)erpendiciilar  to 
it.  Then  ON  is  the  virtual  velocity  of 
P,  and  P .  ON  is  the  virtual  moment, 
virtual  displacement  of  the  point. 

If  the  projection  of  the  virtual  displacement  on  the  line 
of  the  force  lies  on  the  side  of  O  toward  which  P  acts,  as  in 
the  figure,  the  virtual  velocity  i  •  considered  positive;  but 
if  it  lies  on  the  opposite  side,  /.  e.,  on  the  action  line  pn^ 
longed  throngh  0,  it  is  neyative.  The  forces  are  always 
regarded  as  positive  ;  the  sign,  therefore,  of  a  virtual  mo- 
ment will  bo  the  same  as  that  of  the  virtual  velocity. 

CoK.— If  0  be  the  angle  between  the  force  and  the  virtual 
displacement,  we  have  for  the  virtual  moment, 

P  .  ON  =  P  .  OA  cos  0  =  P  cos  0  '  OA. 


OA  is  called  the 


♦  Tlip  priiii'iplo  of  Viiliml  Vclodtios  wan  lil^<-^lverl■(l  by  Oallleo,  and  waH  very 
fully  (Icvclopod  by  Bcrnoiiilll  mill  LaKniii^'c. 

t  SdinoiimcH  culled  "  Vlrluul  Work."    'I'ljc  iiaiuo '■  Virtual  Moment"  \va»  (ilveii 
by  Diilimnel. 


DCITIES* 

■  applicafion  of 
definilely  fonall 
t  the  direction 
the  force ;  and 
locify  has  been 


N 
Fig.  50 


.  is  called  the 

it  on  the  line 
h  P  acts,  as  in 
positive;  but 
tion  lino  }ir(>- 
;cs  are  always 
a  virtual  mo- 
clocity. 

md  the  virtual 
.  OA. 


illleo,  and  waH  very 


VIRTl  Ah    1  KLOCITIKS. 


liy-i 


Now  P  cos  0  is  the  projection  of  the  force  on  the  direction 
of  the  displacement,  and  is  equal  to  OM,  OP  bein*,'  the 
force  and  PM  being  drawn  perpendicular  to  OA.  Hence 
we  may  also  define  the  virtual  moment  of  a  force  as  the 
product  of  the  virtual  dispiitrcmeiit  of  its  point  of  applica- 
tion into  the  projection  of  I  lie  force  on  the  direction  of  thix 
dixplaccment ;  and  this  definition  for  some  purposes  is 
more  convenient  than  the  former. 

Remark. —A  fore;'  is  said  to  do  work  if  it  moves  the  body  to  which 
it  is  ai)i)lied ;  und  the  work  done  by  it  is  nieiisured  by  the  product  of 
tlie  force  into  tiie  space  through  which  it  moves  the  body.  (Jenerally, 
tiio  work  (Ume  by  any  force  durintr  an  infinitely  small  displacement  of 
its  point  of  application  is  the  product  of  tlie  resolved  part  of  the  force 
in  the  direction  of  the  displacement  into  the  displacement ;  and  this 
is  the  8!ime  as  the  virtudl  moment  of  the  force. 

102.    Principle    of  Virtual    Velocitiep    -(1)    The 

virtual  moment  of  a  force  is  cjual  to  t/ie  sum  /  .'  ■  virtual 
moments  of  its  components. 

Let  OR  represent  a  force.  A',  act- 
.ng  at  0;  and  let  its  components  be 
/^  and  Q,  represented  by  OP  and 
OQ.  r^et  OA  l}e  the  virtual  dis- 
placement of  0,  and  let  its  projec- 
tions on  /'.  P,  and  Q,  he  r,  p,  and 
q,   respectively.     Then    the    virtual 

moments  of  these  forces  are  A'  •  r,  P  •  p,  Q  ■  <]■  I>i'iiw  Jin, 
Pm,  and  Qo.  i)erpendicular  to  OA.  Then  On,  Om,  and  Oo 
{=  nui),  are  the  projections  of  R,  P,  and  Q,  on  the  direc- 
tion of  the  displacement ;  and  lience  (Art.  101,  Cor.)  we  have 

R.r  =^  OAOk; 
P.p=  OA-Om; 
Q  .  q  z=  OA  •  mn. 


Fig.SI 


I(i8 


viiiTV. I L  \hh(icrnES. 


Uencc  P  •/'  ^-  Q  -q  —  <)A  ((>///  -\-  mn) 

—  OA  .  On  =  A'    r. 
(See  Miucliin's  Statics,  p.  08.) 

{'i)  If  t'  eiT  fli(  any  inimbcr  of  component  forces  we  may 
eonipoi.nd  tin  in  in  order,  takiiig  any  two  of  tln'm  first,  and 
finding  the  virtual  moment  of  tlieir  resnltant  an  aitove,  then 
finding  the  \irtual  moment  of  the  resultant  of  these  two 
and  a  third,  likewise  the  virtual  moment  of  the  resultant  of 
the  (irst  three  and  a  fourlli,  and  .^o  on  to  the  last  ;  or  we 
may  use  the  polygon  of  forces  (Art.  .'5IJ).  The  sum  of  the 
virtual  moments  ul  the  forces  is  etpial  to  the  virtual  dis- 
l)laeement  mullipiied  hy  tjie  sum  of  tlie  j)rojeetions  on  the 
displa"oni(iit  of  ihe  sides  of  the  j)olygou  which  represent 
the  forces  (Art.  101,  Cor.).  But  the  sum  of  tiiese  projec- 
tions is  equal  to  the  projection  of  the  remaining  side  of  the 
polygon,*  and  this  side  represents  tlie  resultant,  (Art.  3;], 
Cor.  1).  Therefore,  the  tiiim  of  the  virtual  mumeiits  of  any 
number  of  concurring  forces  is  equal  to  the  virtual  moment 
of  the  resultant. 

{[])  If  the  forces  are  in  e(piiUhrium,  their  resultant  is 
equal  to  zero  ;  hence,  it  follows  that  wtien  any  number  of 
concurring  forces  are  in  equilibriitm,  the  sum  of  their 
virtual  moments  —  0. 

This  princii)le  is  generally  kiu)\vn  as  the  J'rinciple  of 
Virtual  Velocities,  and  is  of  great  use  in  the  solution  of 
practical  problems  in  Statics. 


•  Prom  the  nature  of  projcctionn  (Anal.  Qoom.,  Art.  IBS),  it  If  clear  that  in  any 
oerlew  of  points  the  projection  (on  a  ;:iv(Mi  line)  ofllie  line  wliicli  joinn  Uic  (Irnt  and 
lasl.  is  equal  to  llio  funi  of  llie  project  ions  of  ihe  lines  which  join  llu(  points,  two 
anil  two.  Thus,  if  Ihe  siiies  of  a  cIom'iI  polyi;on,  taken  in  ordir.  lie  muikeil  Willi 
arrows  poinlint;  from  cat-li  vertex  to  the  next  one;  and  if  their  projeelions  he 
nnirked  with  arrows  in  the  same  direelioiis,  then,  lines  measured  IVoui  ieli  to  riulil 
heiiiK  considered  positive,  and  lines  Iroin  ri(!hl  lo  left  negative,  llu  sum  of  the  piv- 
jeetions  qf  (fte  dilcs  of  a  dosed  iioli/con  or.  any  right  Hive  is  zero. 


1 


VIRTUAL   MOMENTS. 


1G9 


"0 


II (  forces  we  may 
jf  Hi<'ni  first,  and 
lilt  as  al)ove,  then 
aiit  (if  tlu'se  two 
f  tlie  rL'siiltunt  of 
tlie  lust ;  or  we 
The  sum  of  the 
tiio  virtual  dis- 

I'ojc'ctioiis  oil  the 
wliicli  represent 
of  tiiese  projec- 

lining  side  of  the 
ultaiit,  (Art.  ;5;j, 

I  moments  of  any 

e  virtual  moinoit 


iieir  resultant  is 
1  any  iiiinidtr  of 
he  sum   of  thvir 


:\\{i  PrincipU  of 
the  solution  of 


,  il  in  clciir  Hint  in  uny 
liicli  joiiiB  I  he  flrHt  and 
•Il  join  Hie  |Hiiiil«.  tv.d 
(irilcr.  Ill'  in.nkcd  willi 

If    llll'il'    |I1'I||('I'||<I||H    III' 

'Utcd  lidiii  Idl  111  riu'lil 
M',  llu  sum  of  the  inv- 

IV. 


103.  Nature  cf  the  Displacement. — It  must  he  care- 
fully ohserved  tiiat  the  di8i)laeeiiieiit  of  the  ])artiele  on 
which  the  forces  act  is  virtiiul  and  arbitrary.  The  word 
virtual  in  Statics  i'i  used  to  intimate  that  the  displacements 
arc  not  really  made,  hut  only  supposed,  /.  e.,  they  are  not 
actual  l)iit  imagined  displacements;  hut  in  the  motion  of  a 
jiarticle  treated  of  in  Kinetics,  the  displacement  is  often 
taken  to  he  that  which  the  jiarticle  actually  undergoes. 
Ill  Art.  101.  the  displacement  was  hmited  to  an  iniinitesi- 
mal.  In  some  cases,  however,  a  finite  displacement  may  he 
used,  and  it  may  h  "i^ven  more  convenient  to  consider  a 
finite  displacement.  lint  in  very  many  cases  any  finite  dis- 
placement is  sullicient  to  alter  the  amount  or  direction  of 
the  forces,  so  as  to  prevent  tiie  jirinciple  of  virtual  velocities 
from  heing  apj)lical)le.  This  dilliculty  can  always  he  avoid- 
ed in  practice  hy  assuming  the  displacement  to  he  intinitesi- 
mal  ;  and  if  the  virtual  displacement  is  infinitesimal  the 
virtual  velocities  are  all  inlinitesimal. 

104.  Equation  of  Virtual  Moments.— Let  1\,  P„, 
7*3,  etc.,  denote  the  forces,  and  6p^,  6p.^,  i^p.^,  etc.,  the  vir- 
tual velocities  ;  then  the  jirinciple  of  virtual  velocities  is 
expressed  (Art.  102)  hy  the  o(piation 

J\  '  ^Pi  +  ^^\  •  'h'-i  +  ^'i  '  '^J>3  -r  etc.  =  0; 
or  I-PSp  =  0,  (1) 

wliich  is  called  the  equation  of  virtual  monmiis.* 

Sen. — If  the  virtual  displacement  is  at  right  angles  to 
the  direction  of  any  force,  it  is  clear  that  dp,  the  virtual 
velocity,  is  eipial  to  zero.  Ilciive,  7rhen  the  virtual  tlis- 
placement  is  at  rii/hi  unyles  to  tlie  direction  of  tlie  force. 


*  Or  WWwni  M'ort  (Sec  Art.  101,  Rom.).  Thin  eqimtioii  liu«  been  made  by  I.a- 
Kmiii;e  the  fmindatlon  of  bin  great  work  on  MecUanlCM,  "  Mocaniqut  Analytiqiie." 
(PriQcV  Anal.  Mecb.,  Vol.  I,  p.  US.) 


no 


Sl'STHM   OF   I'AHTIVLKS. 


the  virtual  inuineiU  vf  the  force  =  0,  and  t tie  force  will  not 
enter  into  tlie  equation  of  cirtual  moments. 

Such  a  virtual  (lisi)laccuiout  i«  always  u,  convenient  one 
to  c'hoot^e  wlicn  we  wish  to  get  rid  of  some  unknown  force 
wiiieli  acts  ujjon  a  particle  or  system. 

105.  System  of  Particles  Rigidly  Connected.— (1) 

If  a  i)artiele  in  cciuilibrium,  under  tiie  action  of  any  forces, 
he  constrained  to  maintain  a  fixed  distance  from  a  given 
fixed  point,  tlie  force  due  to  the  constraint  (if  any)  is 
directed  towards  the  fixed  point. 

Tjet  W  he  the  particle,  and  A  the  fixed  ]ioint.  Then  it  is 
clear  tliat  w .  may  substitute  for  the  string  or  rigid  rod 
wiiicii  connects  B  witli  A,  a  smooth  circular  tube  enclosing 
the  i)article.  witii  the  centre  of  the  tube  at  A.  Now,  in 
jrder  tliat  B  may  be  in  equilibrium  inside  the  tube,  it  is 
necessary  that  the  resultant  of  the  forces  acting  upon  it 
should  be  normal  to  the  tube,  /.  e.,  directed  towards  A. 

(2)  liet  there  be  any  number  of 
particles,  vi^,  vi.^.  m^,  etc.,  each 
acted  on  by  any  forces,  Pj,  Pg,  Pj, 
etc.,  and  connected  with  the  others 
by  inflexible  right  lines  so  that  the 
figure  of  the  system  is  invariable. 
Then  each  particle  is  acted  on  by  all 
the  ixtcrnul  forces  ap])lied  to  it,  and 
by  all  the  iiifprndl  forces  ])roeeeding  from  the  internal  con- 
nections of  the  particle  with  the  other  ])articles  of  the 
system.  Tints  tlie  j)artiele,  m.  is  acted  on  by  P,,  /'g,  etc., 
imd  by  tlie  internal  forces  whieii  proceed  from  its  con  nee- 
lion  will)  //<,,  //(g,  V//3.  etc..  and  which  act  along  the  lines. 
Duii^.  iiii>i.^.  etc.,  by  (1)  of  this  Article.  Denote  the  forces 
•dong  the  lines  iiiiiii,  nini.^,  iiiiii.^.  etc.,  by  t^,  t^,  t.,,  etc.. 
.aid   their   virtual   velocities   by  (J/j,  fVj,  dYg,  etc.      Now 


} 


'le  force  ivill  not 


convenient  one 
unknown  force 


lonuected.— (1) 

n  of  any  forces, 
!e  from  ii  given 
■aint  (if  any)  is 

int.  Tiien  it  is 
ng,  or  rigid  rod 

■  tube  enclosing 

i't  A,  Now,  in 
lo  the  tube,  it  is 
I  acting  upon  it 

towards  A. 


lie  internal  con- 
lartioles  of  tlie 
).vP,,  /',,,ete., 
•oni  its  eoniice- 
ilong  tiie  lirie.-i. 
note  Hh'  forces 
/j,  /g,  I.,,  etc.. 
/g,  etc.      Now 


SYSTEM   OF  PAKTICLKS. 


171 


iniaijine  tliat  the  system  is  >liglitly  disjjlaccd  so  as  to 
occui)y  •''  "^^^'   i>osition.      Then   (1)  of  Art.    104:  gives  us 

foi'    111, 

I\6p^  +  l^Sp^  +  etc.  +  /,(5/,  +  /,<5/3  +  etc.  ^0,  (1) 
for  nil, 

^V/h  +  A'Vs  +  etc.  +  /,'5/i  +  //^2  +  etc.  =  0,    (-J) 

proceeding  in  this  way  as  many  ecpuitions  may  be  formed  as 
tiiere  arc  particles  in  the  system. 

Now  it  is  clear  that  /j'V,,  and  L^Sf^,  in  (1)  have  contrary 
signs  from  what  they  have  in  (2).  Thus  if  the  system  is 
moved  to  the  rii/fi/  in  its  disphicement,  /"I'Vi,  and  /z'^/.^  will 
be  ])ositive  in  (I)  and  negative  in  {'i)  (Art.  lUl),  and  hence, 
if  we  add  (1)  and  (•■.')  together,  these  terms  will  disappear; 
in  tlie  same  way,  the  virtual  moment  of  the  internal  force 
along  the  line  connecting  /ii  with  any  other  ]>article  disap- 
pears by  addition,  and  the  same  is  true  for  the  internal 
force  between  any  two  particles  of  tlic  system.  Hence, 
adding  together  all  the  equations,  the  internal  forces 
disappear,  and  the  resulting  etpiation  for  the  whole 
system  is 

^P6p  =  0,  (1) 

and  the  same  result  is  evidently  true  whatever  be  Mie  num- 
i)er  of  particles  forming  the  system.  Hence,  {f  nmj  iiinii- 
ber  of  forces  in  a  xi/sfem  arc.  in  equilibrium,  the  iiim  of 
their  virtual  moments  ^^  0. 

The  converse  is  evidently  trnc,  that  if  the  sum  of  the 
virtual  moments  of  the  forces  vanishes  for  every  virtual 
displacement,  tlie  system  is  in  ef|uilibrium. 

The  following  are  examples  which  are  solved  by  the 
principle  of  virtual  velocities. 


17a 


KXAMVLES. 


EXAMPLES. 

1.  Determine  tlie  condition  of  fqiiiiihriuin  of  u  heavy 
body  resting  on  II  smoulli  inclineil  i)lune  nnder  tlie  action 
of  given  forces. 

Let  \V  be  the  weiglit  of  tlie  body 
sustained  on  the  plane  BC  by  tlic 
force,  r,  making  an  angle,  0.  witli  the 
plane.  To  avoid  bringing  the  un- 
known reaction.  R.  into  our  equation, 
Ave  make  the  displacement  of  its  point. 
ol'apj)lication  ])eri)endicular  to  its  line  of  action,  (Art.  104, 
Sell.);  hence  \vc  conceive  O  as  receiving  a  virtual  dis- 
placement, OA,  at  right  angles  to  \\,  the  magiiitiide  of 
which  in  the  present  case  is  unlir.ited.  Draw  \}ii  and 
Am  jK'rpeiidicular  to  \V  and  P  resjiectivcly,  Om  and  ()// 
are  the  virtual  velocities  of  W  and  1*,  (Art.  101)  ;  and 
W  •  0//t  and  1*  •  0;/  are  their  virtual  moments.  Hence  (1) 
of  xVrt.  10-i,  gives 

W  .  0)11  -  V  ■  On  =  0. 
But  Om  =  Ox\  sin  «, 

and  On  =  OA  cos  d ; 

therefore  W  sin  «  —  1^  cos  f)  --  0;  (1) 


which  airrees  with  Ex.  '■>,  Art.  41. 


If      •<>   force  acts   parallel   lo    I  lie  plane.  0 
becomes 

V  =  W  sin  «; 


0,  and  (1) 


which  agrees  with  lv\.  i,  Art.  41. 


' 


rill  111  of  a  lieav)' 
under  tliu  act  ion 


.R 

;  c 

~k^^ 

\^ 

iw 

Fig.53 

action,  (Art. 

104, 

iiX   a   virtual 

dis- 

lie   nia<,Miitn(i 

0  of 

Praw  \ii} 

and 

M'lv,  0;/(  and 

()// 

(Art.   101); 

and 

cuts,     llonco 

(1) 

(1) 


0  =  0,  and  ( 1 ) 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


I 
■I 


1.0 


I.I 


2.2 


1^ 


1.8 


1.25      1.4 

1.6 

^ 6"     

► 

s 

3 


I 


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KXAMI-Lh:S. 


\r.) 


2.  Suppose  tho  pliiiio  in  Ex.  1  t'>  '»<>  rniigli,  and  tlnit  the 
body  is  on  the  point  of  ix'ing  dragged  \ip  the  phuie,  lind 
tlie  condition  of  eipiilihriitni. 

Tlie  nonnal  resistance  wdl  now  he  '^'"1    a/  ^-p 

replaced  by  tlie  total  resistance,  IJ,         "^^ xJK?^^^^^^ 
inclini'd  to  the  normal  at   an  angle  ^^^'o" 

~-  </),  the  angle  of  friction  (Art.  i)5,      ^^ili — __ 

Cor.).     TiCt  the  virtual  displacement,  Fia.54 

OA,   take   place   i)erpendicularly  to 
11,  then  (1)  of  Art.  104,  gives 

W  •  Oin  -  r  ■  0?i.  =  0. 

Hut  Om  =  O.V  sin  («  +  (p), 

iind  0«  =  O.Vcos(<A-^0; 

therefore        W  sin  («  +  (p)  =  P  (•<)s  (r/)  —  0) ; 

which  agrees  with  (15)  of  Art.  DG. 

;}.  Determine  the  horizontal  force 
which  will  keep  a  particle  in  a  given 
])()sition  inside  a  circnlar  tube,  (1) 
when  the  tube  is  smooth  aiul  {'l) 
when  it  is  rough. 

(1)  Let  the  virtual  displacement, 
OA,  be  an  infinitesimal,  =  ilx,  along 
the  tube.     'Phen  since  ds  is  intinites- 

iinal  the  virtual  velocity  of  U  =  0.     Then  the  equation  of 
virtual  moments  is 

_  W.0»(  +  V-ihi  =  0. 


Fig. 55 


lint 

and 

liicrefore 

or 


Om  =  (Is  ■  sin  0, 
0)1  =  i/s  •  cos  0  ; 
\y  .  sin  «  =  !'•  <os«; 

1>   rr:    W  tall  <>. 


174 


KXAMI'Llu'i. 


(2)  Siipi)oso  tlio  force,  P.  just  sustains  tlic  particle  ;  the 
iiurmal  resistance  must  now  he  replaced  by  the  total  resist- 
ance, nnikiujf  the  anirle,  <p,  witii  tiie  normal  at  the  ri<riit  of 
it.  Take  tiie  virtual  displacement.  OA'.  at  right  angles  to 
tiie  total  resistance  (Arh  105.  Seh.),  and  let  it  l)e  as  before, 
an  infinitesimal  dn.     Then  (1)  of  Art.  lU-i,  gives 

—  W  •  Oct  -I-  P  •  On'  =  0. 

But  Om  =  (Is  •  win  (0  —  <p), 

and  On'  —  ds  •  cos  {6  —  <j>), 

therefore      W  •  sin  {0  —  <l>)  =  F  -  cos  {0  —  (f)); 

or  P  =  W  .  tan  (9  —  0). 

Similarly,  if  the   force,  P,  will  y!<.«i'  draff  the  particle  up 
the  tube  wo  obtain 


P  =  W  .  tan  (0  +  0). 

4.  Solve  by  virtual  velocities  Ex.  G,  Art.  03. 

Let  the  displacement  be  made  by  diminishing  the  angle 
«,  which  the  beam  makes  with  the  horizontal  plane,  by  dn, 
the  ends  of  the  beam  still  remaining  in  contact  with  the 
horizontal  and  vertical  pla'ies.     Then  I  he  virtual  velocity  of 


T  =  rf  •  2rt  cos  fc  =  —  2a  sin  «  da; 


and  that  of 


W  =  f?rt  sin  «  =  rt  cos  «  da, 


and  those  of  the  reactions.  R  and  R',  vanish.     Then  tiie 
e(puition  of  virtual  moments  is 


mmm 


JO  particle  ;  tlie 
the  total  resist - 

I  at  the  ri^'iit  of 
right  angles  to 
it  l)e  as  before, 


0); 


the  particle  up 


03. 

ishing  the  angle 
al  plane,  by  dn, 
'ontact  with  the 
irtual  velocity  of 

t  (la; 


lish.     Then  the 


^ 


EXAMPLES.  1T5 

—  T  2a  sin  «  ila  +  W  n  cos  «  <Ui  —  0 ; 
.•.     -i'V  sill  a  —  W  cos  a. 

5.  Solve  Ex.  8,  Art.  (12,  l)y  virtual  velocities. 

Let  the  displacement  he  made  by  increasing  the  angle 
0  by  (10,  the  point,  A,  remaining  in  contact  with  the  wall; 
the  virtual  displacement  of  B  is  at  right  angles  to  the 
direction  of  the  tension,  T-  and  hence  the  virtual  moment 
of  T  is  zero  ;  the  virtual  velocity  of  W  is 

d{b  cos  (p  —  a  cos  0)  z=  a  sin  0  dO  —  h  sin  ^  dip. 
Then  (1)  of  Art.  104,  gives 

W  {a  sin  6  dd  —  h  sin  (/>  d(p)  =  0; 
.*.     b  sin  (f)  dip  =  a  sin  0  dd. 
But  from  the  geometry  of  the  figure  we  havo 
b  sin  0  =  2a  sin  0; 
,•.    b  cos  0  dcp  =■  2a  cos  6  dd; 
.'.    2  tan  0  =  tan  ft; 

which,  combined  with  ("))  of  Ex.  8,  Art.. 62,  gives  us  the 
values  of  sin  0  and  cos  </> ;  and  these  in  (0)  of  that  Ex. 
give  us  the  value  of  x. 

G.  Solve  Ex.  38,  Art.  65,  by  virtual  velocities. 

Since  tin-  bar  w  to  rest  in  all  positioiiH  (in  t}io  curve  and  tlie 
jK'f;,  its  centre  of  gravity  will  neither  rise  nor  fall  whe.i.  tlio 
bur  receivet:  a  displacenient,  lienco  its  virtual  velocity  wi'l— 0; 
.  • .    etc. 


EXAMPLES. 


T 


7.  In  Ex.  4,  Art.  4'-3.  prove  tliat  (1)  is  tlic  equation  of 
virtual  nionients. 

S.  Find  the  incliniition  of  the  l)eain  to  the  vertieal  in 
Ex.  ;il,  Art.  tl5,  by  virtual  velocities. 

1).  Deduce,  by  virtual  velocities,  (1)  the  formula  for  the 
triangle  of  forces  (see  1  of  Art.  32),  and  ("J)  the  formula  for 
the  parade logruui  of  forces  (Sec  1  of  Art.  130). 


^ 


4 


pqiiution  of 

viTticiil  in 

liila  for  tlic 
'onmilii  for 


\ 


C  H  A  P  T  i:  R     VII. 

MACIIIXES, 

lOS.  Functions  of  a  Machine. — .1  niKc/iiiie,  Slaltcauy, 
i.i  ((11 11  tWlCKOW  f  by  iiwaiis  nf  (fhicli  ne  in((y  c/KOiyr  the 
(lircc/icii,  'i)i(i(ii(i!>i'le.  uml  poiuf  af  ((/■/ilicKliitii  nf  k  ijiceii 
fiircc  ;  ((,!'(  h'iiir/i'c.tl/f/,  il  is  any  inslndia'Dl  by  iiic((hh  of 
(t'ltirb  (cv  may  <•!(((  d'ji;  the  (/ir(ctioii  and  cchnily  of  a  yireit 
itintion. 

Ill  !iiii)lyiiif.r  tlic  ]iiin  •i;)lc'  of  virtuiil  vclocilicH  to  ii  system 
of  coiiuectod  bodios,  •!(' viin'iiUL'  is  gaiiit'd  l)y  ciioosii'ig  tiio 
virtual  di.sphu'omoiits  ]«  t'flaiii  directions  (Art.  104,  fScli). 
When  \VL'  use  this  |)rinci|ilt"  in  the  discussion  of  niaeliines 
the  disjilaceiiieiits  wiiicii  we  si. all  ciioose  will  be  those  wliieli 
the  dMl'erent  jiarts  of  a  mavhuw  (((ina/ly  undergo  when  it 
is  eiiii»Ioyed  in  doing  work,  iiiid  instead  of  c(|iiations  of 
virt !(((/  wovk  we  shall  liuvo  e(|iiaiioiis  of  art((al  work;  and 
in  future  the  principle  of  virtual  velocities  will  often  be 
referred  to  as  the  J'rinci/ili'  of  W'or/i'.  (See  Minehin's 
Statics.  )).  :)S;i.) 

Every  niacliiiu'  is  designed  for  the  purpose  of  overcoming 
certain  forces  which  are  called  resi.st(()ii)s ;  and  the  forces 
which  are  applied  to  the  machines  to  produce  this  effect  are 
called  iiwviny  forces.  When  the  niachine  is  in  motion, 
jvery  moving  force  displaces  it >  point  of  application  in  its 
own  direction,  while  the  point  of  applicalion  of  a  resistance 
is  displaced  in  a  direction  opjiosile  to  that  of  the  resistance. 
Hence,  a  moving  force  is  one  whose  elementary  work*  is 
positiec,  and  a  resistance  is  one  whose  elementary  work  is 
ue(intire.     The  moving  forco  is,  for  coiueiiieiice.  called  the 


*  See  All.  101,  Uoin. 


178 


.^f^:< ■//.  i  .\i'  ■.  i  /.  .  i  />  r.  i  xta  h e. 


po/irr ;  iiiul  Ikhmiisc  tlio  iittractioii  of  firuvity  is  tlio  most 
ctMiiiiiKn  i'lU'm  (if  the  force  or  I'csistiiiU'i'  to  lit'  ovt'rcoiiK'  it  is 
usually  ciillt'd  I  lie  tri'lijlil. 

The  wcijrlit  or  resistiina-'  to  lie  ovcrcimic  iiiav  he  tlic  ciirth's  jittrac- 
tioii.  lis  ill  misiiiir  a  wciylit  ;  the  HKilri'iihu-  iittnictiiins  li  twccii  Xhi: 
imrtic'Ics  of  a  liodv  us  in  sti\iii|iiiijj  or  cKiliii!,'  a  metal,  or  dividing 
wood  ;  or  friction,  as  in  drawing  a  licavv  liody  along  a  rough  road. 
Till'  iiiiwcr  iiiay  tie  that  of  iiirii,  or  liorses,  or  tlio  steam  engine,  etc., 
and  may  be  just  suilicieiit  to  overcome  the  resistance,  or  it  may  lie  in 
excess  of  what  is  necessary,  or  it  may  li(>  too  small.  It' just  sufficient, 
the  machine,  if  in  motion,  will  remain  nniforinly  so,  or  if  it  be  at  rest 
it  will  he  on  the  ]ioint  of  moving,  and  the  power,  weight,  and  friction 
will  be  in  eiiiiililiriiim.  If  the  power  be  in  excess,  the  machine  will 
be  set  in  motion  and  will  continue  in  accelerated  liiution.  If  tlie 
|xiwer  bo  too  small,  it  will  not  lie  atile  to  move  the  machine  ;  and  if 
it  he  already  in  motion  it  will  gradually  come  to  rest. 

Till'  griieral  pfohlt'iu  with  iv<,'iird  to  luiiohinos  i.s  to  liiul 
tlie  ri'lation  between  tlie  power  iiiid  tlio  weiglit.  Some- 
times it  is  most  conveiiieiit  that  tliis  roliitioii  siioiilil  bo  one 
of  eqiiiiHty,  i.  c,  tliat  the  power  siioiiKl  e(|iial  the  weight. 
Ceiierally,  however,  it  is  most  eotiveiiient  tliat  llie  power 
siioukl  bo  very  dilTereiit  from  the  weiglit.  Tlius.  if  a  man 
has  to  nt't  a  weight  of  one  ton  liaiiging  liy  ii  rope,  it  is  eloar 
that  ho  cannot  do  it  lude.sa  the  meclianictil  contrivance 
provided  enable  him  to  Hf't  the  \ieigiit  Ijy  exorcising  a  pull 
of  very  ntuoh  loss,  say  one  cwt.  Wlieii  the  ])ower  is  much 
smaller  tlian  the  weight,  as  it  is  in  *^'iis  cii.<e,  wliich  is  a 
very  common  one.  tlio  niiiciune  is  said  to  work  at  a  nirrhait- 
iml  advuntatjc.  When,  as  in  some  other  ciisos,  it  is  dcsindile 
that  tlio  power  should  be  groiiter  than  the  weigiit,  there  is 
said  to  be  a  mcr/iaiiirni  (/is/d/rdti/df/c  of  the  machine. 

107.  Mechanical  Advantage.— (1)  I.et  P  and  M'  bo 
the  power  and  weight,  and  p  and  w  their  virtual  velocities 
respectively;  and  let  friction  be  omitted.  Then  from  the 
equation  of  virtual  work  (Art.  ]U4),  we  have 

P       w 


\ 


Pp  -  ir«' 


or     -rr  =■ 


ty  is  t  lie  most 

lUiTCOllK'  it   is 

ic  ciirtli's  jiltrac- 
nils  1)  twccii  tli(? 
('till,  or  dividing 
iig  a  iDiigli  read, 
'iiin  engini',  etc., 
or  it  may  Iw  in 
It'  just  .sufflcit'iit, 
)r  if  it  be  at  rest 
iglit,  and  friction 
he  Duicliine  will 
motion.  If  tlie 
machine  ;  and  if 

lines  i.s  to  liiid 
eifflit.  Somo- 
siioiild  be  one 
111  the  \vei<i;lit. 
lilt  tlie  power 
hns,  if  a  man 
ope,  it  is  olear 
il  eoiitrivaiiee 
reisiiig  a  piil! 
)()\ver  is  much 
so,  which  is  a 
k  at  a  mrrJiau- 
,  it  is  desinihle 
.'ight,  there  is 
achiiie. 

P  and  ir  1)0 
•tual  velocities 
hen  from  the 


.»//,''  7/.  I  .V/'  A  I,    A  U\A  \TA  H  K. 


lilt 


1 


whicli  shows  that  the  smaller  /' is  in  eomparisoii  with  li'. 
l!io  sinidlor  //•  will  lie  in  cotiiparison  witii  ji.  I'lit  I  lie 
smaller  /'  is  in  coinparisoii  with  IT.  the  <rreater  is  the 
tiiirlKtiiiral  (K/i'dii/aiir.  Hence,  the  <;ivaler  the  mechanical 
iidvsinla.ne  is.  the  less  will  lie  the  virtiiiii  velocity  of  the 
woiLrlit  in  coir.j)iiri<on  with  tluit  ol'  'he  powc'r.  Now,  if 
motion  Mclniilly  takes  phuv  thi'  rir/ind  velocities  hoconu' 
iicfiKd  \o\oc\t'w>i\  and  lieneo  we  have  the  i>rineiple  ir/iat  is 
ijainvd  in  pourr  is  lost  in  ir/ori/i/. 

{•■I)  There  are  no  cases  in  which  the  weight  and  power 
arc  the  only  forces  to  he  considered.  In  every  movement 
of  a  machine  there  will  always  he  a  eerttiin  timoniit  of  fric- 
tion; iind  this  can  ne\er  he  omitted  from  the  e(piiitioii  of 
virtual  work.  There  are  eases,  however,  iis  liiat  of  a  biilance 
on  !i  knife-edge,  where  the  friction  is  very  small;  and  for 
these  the  jiriiiciple,  winit  is  gained  in  jiower  is  lost  in 
velocitv,  is  very  approximately  trne.  Where  the  friction  is 
eonsideriil)le  this  is  no  longer  the  ease. 

Let  /'  and  /  he  the  resistance  of  friction  and  its  virtual 
velocity,  then  the  equation  for  any  machine  will  take  the 
form 

Fj)  -  Wir  -  y;/'  =  0, 

which  shows  ns  tlnit  iiltlioiigh  /'ctin  he  made  as  small  as  we 
wish  hy  taking  p  large  enough,  yet  the  mechainciil 
advantage  of  diminishing  /'  is  restricted  hy  the  fact  that  / 
inciviises  witliy^;  ami  therefore  as  /'diminishes  there  is  a 
e  n-ri'sponding  increase  of  the  work  to  he  done  against  fric- 
iion.  Hence  if  friction  bo  neglected,  there  is  no  jn-actical 
Hniit  to  the  ratio  of  P  to  II';  bnt  if  the  friction  be  con- 
siilered.  the  advaiitiige  of  diminishing  /'  has  a  limit,  since 
if  Pp  remains  the  snine,  Wir  must  decrease  as  /y  increases; 
(.  c,  the  work  done  against  friction  incroMses  with  the 
comide.xity  of  the  machine  :  and  tiins  jints  a  practic:il  limit 
to  the  mechiinical  ml  vantage  which  it  is  possible  to  obtain 
by  the  use  of  machines. 


ISO 


SLUr  L  E  MA  cmsES. 


108.  Simple  Machines. — 'Plic  simple  iimcl)iiu'.s.  some- 
tinic's  called  the  Mcchanirdl  I'mrrrK,  are  geiiLnil!;.  eiiuiiier- 
ate'l  us  six  in  iiiiuilier  :  tlie  L)'rci\  I  lie  11//*  (7  und  Axir,  the 
Iiidimd  l'iani\  the  /'iillci/,  the  HVv///r.  ami  the  Scrcir. 
Tlie  Lcrcr,  the  Indinnl  /'laiir,  ami  the  Pii/Iri/,  may  he 
considered  a.s  distinct  in  principle,  while  the  othe  .•;  are 
conihinations  of  them. 

The  efficiency*  of  a  machine  is  the  ratio  of  the  useful 
work  it  yields  to  the  whole  amount  of  work  ])erf()rmed  hy 
it.  The  UHeful  work  is  tha(  whit-h  is  performed  in  over- 
coming useful  resistances,  while  lust  work  is  that  which  is 
spent  in  overcoming  waslefid  resistances.  Usefnl  resist- 
ances are  those  which  the  machine  is  specially  designed  to 
overcome,  while  tho  overcoming  of  icastefiil  resistances  is 
ftjreign  to  its  purpose.  Fric/ioii  ami  ri(/i(/i/>/  of  cords  are 
wftsteful  resistances  while  the  weiijlit  of  the  l)iidy  to  ho 
Jifted  is  the  useful  resistance. 

liCt  W  he  the  work  done  hy  the  moving  forces,  H'y  the 
useful  and  Wi  the  lost  work  when  the  machine  is  moving 
uniformly.     Then 

W  =  \Vu  +  Wi, 
and  if  .1/  denote  the  etiicieiicv  of  the  machine,  we  liave 


M  = 


Wu 

w' 


In  a  perfect  m.nchine,  where  there  is  no  lost  work,  the 
efficiency  is  unity;  hut  in  every  machine  some  of  the  work 
is  lost  in  overcoming  wasteful  resistances,  so  that  the 
efficiency  is  always  less  than  unity:  and  the  object  of  all 
improvements  in  a  machine  is  to  bring  its  eflieiency  as  near 
unity  as  possible. 

The  most  noticeable  of  the  wasteful  resistances  are  fric- 
ti(m  and  rigiditv  of  cords  :  and  of  these  we  shall  consider 


*  Sometiiiios  called  iiutUulm. 


EQVlLlBltirM   OF  THE  LEVEK. 


181 


iiiicliiiu',«,  some- 
iitnil!;.  I'liiiiiH'i-- 

■/  (lllil  Axir.    ill,. 

ml  tlic  Scrvir. 
'nUvij,  in:iy  lie 
tliL'    ollic  s    arc 

0  of  the  Hifcful 
k  jiort'ormod  l)v 
lurineil  in  ovof. 
.s  tliat  w I)  it'll  is 
Vsvfid  ivsist- 
illy  desiijnod  to 
(I  resistances  is 
'////  o/ninls  arc 
the  hody  to  ho 

;•  forces.    \\\  (]io 
^hiiie  is  niovinir 


only  the  lirst.     Tlio  student  who  wants  information  on  tht 
A  experimental   laws  of  the  rii^'idity  of   cords    is   referred   to 

\Veisl)acirs  Mechanics,  Vol.  1,  p.  :j(i:j. 

109.  The  Lever. — A  lever  is  a  rii,nd  liar,  straight  or 
curved,  niovahle  ahout  a  lixi'd  axis,  which  is  called  the 
I'lili'ruin.  The  parts  of  the  lever  into  which  the  fulcruni 
divides  il  are  called  the  (irin!<  of  the  lever.  When  the  arms 
are  in  a  straight  line  it  is  called  a  strdiijlii  lever ;  in  all 
other  ca.^es  it  is  a  benl  lerer. 

Levers  are  divided,  for  convenience,  into  three  kinds, 
according  lo  the  })Osition  of  the  fiilcriiin.  In  the  first  kind 
the  riilcriim  is  hetwecn  the  power  and  the  weight  ;  iii  the 
.secon<l  kind  the  weigjit  ),cts  between  the  fiilcrnm  and  the 
jiower  ;  in  the  third  kind  the  power  acts  between  the  fn' 
criim  and  the  weight.  In  the  last  kind  the  power  is  always 
greater  than  the  weight. 

A  pair  of  scissors  furnishes  an  examjile  of  a  pair  of  lovers 
of  the  lirst  kind  ;  a  pair  of  nut-crackers  of  tiio  second  kind; 
and  a  pair  of , shears  of  the  lliird  kind. 


le,  wo  have 


lost  work,  the 
ne  of  the  work 
■.  so  that  the 
he  oiiject  of  all 
icieucy  as  near 

dances  are  fric- 
.'fhall  consider 


110.  Conditions  of  Equilibrium 
of  the  Lever. — (1)  Wdlunit  l-'rielmn. 
lict  AB  be  the  lever  and  V  its  fulcrum; 
and  let  the  two  forces.  /'  and  IT,  act  in 
the  plane  of  the  paper  at  the  jioints,  A 
and  H.  in  the  directions,  AI'  and  BW. 
From  C  draw  CD  and  CR  perpendicular  to  the  directions 
of  /'  and  ir.  Let  a  and  (i  denote  the  angles  which  the 
directions  of  the  forces  make  with  the  lover.  Then,  taking 
moments  around  C,  we  liavo 


yr 


P.CD  r=  ir.CE, 

por))endicular  on  direction  of  W 
perpendicular  on  direction  of  /' 


(1) 


182 


EQl'ILiniill'M   OF  THE   LKVKR. 


That  is,  the  condition  of  e({iiilil)riiim  ivqiiires  Unit  (hr 
power  and  wriijld  slmuUt  be  to  each  vther  innrsely  an  Hie 
lenijlk  of  lltcir  rcnpcctice  arm  ft  (Art.  4«). 

To  liud  tlio  pressiiro  on  the  fiilcriiin,  and  its  direction  ; 
let  tl,o  directions  of  the  pressures,  /■•  uiid  H',  intersect  in 
F;  join  C  and  F;  tlien,  since  tlie  lever  is  in  eqiiilihrinni 
l)y  the  action  of  tiio  forces,  P  and  11'.  and  tli"  reaction  of 
tlie  fulcrum,  the  resultant  of  /'  and  W  must  ho  e(|U  d  and 
op|)osito  to  that  reaeticm,  and  hence  must  pass  liu-ough  (,' 
and  he  equal  to  the  pressure  on  the  fulcrinn.  Denote  this 
resultant  hy  R,  the  anjrlo  whicli  it  makes  with  the  lever  by 
Q;  and  the  angle  AFB  hy  w ;  then  we  have  by  (1)  of  Art.  ;50 

IP=:  pi  +  ira  +  X>/'ircos  AFB  ; 

or  li^  =  ya  +  II  2  4-  '>/^ircos  u>,  (a) 

which  (fives  the  pressure,  R,  on  thefiilrrum. 

To  find  its  direction  resolve  /'.  IT,  and  A'  parallel  and 
perpendicular  to  tlie  lover,  and  we  have 

for  parallel  forces,  P  cos  «  -  IT  cos  (i~  U  cos  0  =  0\ 

for  perpendicular  forces,  /'  sin  «  +  W  sin  (i—R  sin  6  =  0; 

by  transposition  and  division  we  get 

/'  sin  «  +  W  sin  ft 


tan  6  =  „ 

P  cos  <(  —  W  i-osfi' 

which  gives  the  direction  of  the  pressure. 


(3) 


Con.— When  the  lever  is  bent  or  curveil  the  condition  of 
eiiuilibriuin  is  the  same. 

Sohdion  try  tlie  principle  of  rirtual  rr/o.ities. 

Supi)ose  the  lever  to  be  turned  round  {.'  in  the  direction 
of  /'  (hrough  the  angle  dd,  into  the  position  at);  let  p  and 


luires  lliat  thr 
Hi'cfscly  as  Ihv 

its  (liroc'tioii  ; 
r,  iiitonst'ct  ill 
in  er|uilil)riiini 
li'»  reaction  of 

be  cc|ii  il  and 

ass  liirongli  (' 

Denote  tiiis 

Il  tlie  lever  by 

■(l)of  Art.lJO 


(2) 


i  parallel  and 

-  A'  cos  0  =  0; 
-R  sin  0  —  0; 

(3) 

!  condition  of 


the  direction 
ih\  let  p  and 


EqCILIBRIl'M  OF  THE  LEVER. 


183 


'/  be  the  perpendiculars  CD  and  CE  respectively,  then  the 
virtual  velocity  of  P  will  be  (Art.  101), 

ka  sin  rt  —  AU-fW-sin  «  =  pdd. 

Similarly,  tV.e  virtual  velocity       II  is  —  qdd. 
Hence,  by  viie  equation  of  virtual  work  we  have 

p. p. lie-  W-q-de  =  0; 
.-.    P-p  =  W-q.  (4) 

which  is  the  same  as  (1). 

(:2)  With  Friction.— In  the  ahovc  we  have  supposed  fric- 
tion to  be  neglected  ;  and  if  the  lever  turns  round  a  sharp 
edge,  like  the  scale  beam  of  a  balance,  the  friction  will  be 
exceedingly  small.  Levers,  however,  usually  consist  of  fiat 
oars,  tiLTuing  aliout  rounded  pins  or  studs  which  form  the 
fulcrums,  and  between  the  lever  and  the  {lin  there  will  of 
course  be  friction.  To  lind  the  friction  let  /•  be  the  radius 
of  the  pin  round  which  the  lover/turns ;  then  the  friction 
on  the  pin,  acting  tangent ially  to  the  surface  of  the  pin 
and  opposing  motion,  =  R  sin  <)>  (Art.  09) ;  and  the  virtual 
velocity  of  the  point  of  application  of  the  friction  =  rdO ; 
and  hence  the  virtual  work  of  the  friction  =  !l  sin  ^-rde. 
Hence  the  ecjuation  of  virtual  work  is 

p.pde  —  W-qdd  —  72  sin  0  rdd  =  0. 

Substituting  the  value  of  R  from  (2),  and  omitting  dO,  we 

have  

Pp  _  117/  =  r  sin  </>  ^pa  +1?^+  2Pircos  w  ;     (5) 

solving  this  quadratic  for  P  we  have 


184 


TUK   COJIMOX  HMjASVE. 


r  =  w 


pq  +  ?•*  cos  w  sin^  0 


III-     •    ^  ^/''  +  '^P'l  '^■'*''  '■'  +  '1^  —  >''~  ^'"'  •/•  ■'^'"^  '•'  /,.i 

/>-  —  /■-  .sui-  </)  ^  ' 

which  pi\('s  tlic  relation  hetwecii  the  jiowrr  aiul  tlic  weight 
wlieii  liietioii  is  eoiisiilered,  the  upper  or  lower  .sign  of 
/■.sill  (p  l)eiiig  lakeii  ueconling  us  /-•  or  IT  is  about  to  pre- 
ponderate. 

('()!!. — If  the  friction  is  so  small  that  it  may  be  omitted, 
r  sill  (p  =  Q,  and  (tj)  becomes 


^  =  ?. 

IF      p 


(7) 


111.  The  Common  Balance. --In  luacliines  geuoraliy 
the  ol)ject  i.s  to  produce  nioiion,  not  rest;  in  other  words 
to  do  work.  The  statical  imestigation  shows  only  the  limit 
of  force  to  lie  api)lied  to  pul  the  machine  on  the  /)ni)it  of 
motion,  or  to  gi\i'  it  'nii/hnii  motion.  For  aip'work  to  he 
done,  the  force  applied  must  I'xceed  this  limit,  and  the 
greater  the  excess,  the  greater  tiu' amount  of  work  dune. 
Tiiere  i.s,  however,  one  cla.ss  of  applicatioiLS  ul'  the  lever 
wliere  the  oiijeet  is  not  to  do  work,  but  tu  jirodnce  e(|ui- 
librium.  and  which  are  therefore  specially  adapted  for  treat- 
ment liy  statics,  'i'jiis  is  the  class  of  measuring  macliiues, 
where  the  object  is  not  to  overc(une  a  particular  resistance, 
but  to  measure  its  amount.  The  testing  maehiue  is  a  good 
example,  measuring  the  pull  wliici;  a  liar  of  any  materi:il 
will  sustain  before  breaking.  The  cimimon  balaiuv  and 
steelyard  for  weighing,  are  familiar  example-. 

The  common  balance  i>  an  instrument  f<u' weighing  :  it 
is  a  le\er  <>['  th^'  tlrsl  kind,  with  two  eipiiil  arms,  with  a 
scale-|ian  suspendi'd  fiMin  each  cNtremily,  the  fulcrum 
being  vertically  :diove  the  ce:itre  nf  Liravity  of  the  beam 
when  the  latter  i>  horizontal,  and  Iherelore  vertically  above 


''J 


THE    fY>.»rVO.V   BALANCK. 


Uit 


p  siii^  (j) 


A'') 


tllC  Wt'lgllt 

or   jsigii    of 
Jilt  to  piv- 


)e  omitiod, 


(7) 


;•?  froiiorally 
thor  words 
ly  llio  liiiiil 
le  /mint  of 
woi'ls  to  he 
it,  iiiid  tlio 
rtoi'k  done, 
f  till'  lever 
idiiee  e(|iii- 
'd  for  freat- 
niiicliiiies, 
resist  anee, 
10  is  a  u'ood 
ny  Tiiateri:il 
lalanee  and 

oiirliiiiu' :  il 
mis,  willi  a 
e  fiilernm 
f  the  heain 
ieally  ahove 


the  eentre  of  gravity  of  the  system  formed  by  the  beam,  the 
.seale-paiis.  and  the  weifihts  of  the  seule-paiis.  The  siib- 
stauee  to  lie  weighed  is  plaeed  in  one  seale-paii,  and  weights 
of  known  inagiiitude  are  jilaeod  in  the  other  till  the  beam 
remains  in  i'(iuilibrium  in  a  jierfectly  horizontal  position, 
in  wliieli  case  1  lie  weight  of  the  substance  is  indicated  by 
the  weiglils  which  balance  it.  tf  these  weights  differ  by 
ever  so  little  the  liorizontality  of  the  beam  will  bo  disturbed, 
and  affor  oscillating  for  a  short  time,  in  oousoquenco  of 
the  fulcrum  lieing  placed  ahore  the  centre  of  gravity  of  tlie 
svsteni.  it  will  rest  in  a  position  inclined  to  the  horizon  at 
an  angle,  tlu'  extent  of  which  is  a  measure  of  the  sensibility 
of  the  balance. 

Tho  prccodinsj:  i-xpliinntioii  rpprcsents  the  balanop  in  its  simplest 
form;  in  practice  tlieic  lire  nuiiiy  niodificiitioiis  and  contiivancea 
intrixhiced.  Mudi  skill  lias  been  ex])enile(l  npon  the  ('(instniction  ol 
balances,  anil  ;i;roat  delicacy  lias  l)een  obtained.  'I'liiis,  the  beam 
fliiinld  !)(■  suspended  by  means  of  a  knit'cedge,  /.  c,  ii  projecting 
metallic  ed-,'!!  tiansver>,e  to  its  lenirtli,  which  rests  ujion  a  plate  of 
agate  or  other  hard  siil)stance.  The  chains  which  support  the  scale- 
pans  should  be  suspended  from  the  extreniities  of  the  beam  in  tho 
same  manner.  Thi'  point  of  supi)ort  of  the  beam  (fulcrum)  should  be 
at  eipial  distances  from  the  i)oints  of  suspension  of  the  scales;  and 
when  the  balance  is  not  loaded  the  l)eam  sliould  be  hoii'.onta).  We 
can  nscer:aiM  if  these  condiliiuis  are  satislied  l)y  ol)serving  whether 
there  is  still  e(|uilibiium  when  the  substance  is  transferred  to  tho 
scale  which  tho  weight  originally  occiiiiii'd  and  the  weight  to  that 
which  the  substance  originally  occupied. 

The  chief  requisites  of  a  good  balance  arc  ; 

(1)  When  eipial  weights  are  ])laoed  in  the  scale-pans  tho 
beam  should  be  perfectly  horizontal. 

(•-')  'I'lie  baiani'c  should  possess  groat  si'iisi/iili/ff  :  i.  p.,  if 
two  weights  which  iirc  very  nearly  oipial  be  plticed  in  tho 
scale  pans,  tho  beam  .should  vary  .^cnsiOIi/  from  iis  horizontal 
position. 


I8fi 


REQUISITES   <iF  A    (lOOD    IIAIjAXCE. 


(;{)   When    tlie    biilaiicc    is    (listurbi'tl    it    slniiild   rt.urily 
ivturii  to  its  state  of  rosL,  or  it  siiould  iiiivo  stabthty. 

112.  To  Determine  the 
Chief  Requisites  of  a  Good 
Balance. — Let  /-•  and  11  lie 
tile  weigiits  in  the  sfak'-paiis  ; 
U  tile  I'ldcniin  ;  //  its  ilistaiice 
from  the  straight  hue,  AH, 
wiiicii  joins  tile  [loiiits  of  at- 
aeluiient   of  tiie   seaic-pans  to 

liie  l)eani;  (J  tiie  centre  of  gravity  of  the  beam  :  and  let 
AH  be  at  right  angles  to  OC.  the  line  joining  tiie  fiilenim 
to  the  centre  of  gravity  of  the  I)eani.  Let  A(!  =r  CM  =t  ,7; 
0(i  =  k;  w  =  tiie  weigjit  of  \\w  lieani  ;  and  0  =  \\w 
angle  whieli  the  l)eani  makes  with  tiie  horizon  when  there 
is  e(|iiilibriinn. 
Mow  the  perpendicnlar  from  0 

on  the  direction  of  /'  =  a  cos  fl  —  //  sin  fl; 
"     "  "  ir  =  </  cos  0  f  //  sin  <9; 

"     ''  "  /^  =  ^-sln«; 

therefore  taking  moinent.s  round  ()  we  have 

P  [a  cos  e~h  sin  0)—  \V  {„  cos  0-\-/i  sin  ti)  —  wk  sin  0  =  0; 


,       ^  (P  -  W)  a 


(1) 


This  c(|nation  determines  the  position  of  e(piilibriiim.    The 
///>/!  req  II  isite— tiie  horizoiitality  of  llie  beam  when    /'and 
If  are  e(pial— is  salisiied  liy  making  the  iirnis  e((iial. 
The  ,s7Y'«//^/ re(piisite  [(">)  of  Art.  Ill],  retpiires  that,  for 

ii  fi'iven  vail f  /'  —  ||',  the  inclination  of  the  beam  lo  ihr 

horizon  must  be  as  great  as  possible,  and  tliercfoiv  ibc  sen- 
sibility is  greater  the  greater  Ian  tl  is  for  a  given  \alue  of 
P  —  H'j  and  for  u  gi\en   \alur  of  tan  M  tlio  sensibility  is 


m 


11  Id   iT.idily 
dity. 


w 


m  :  iiiul  lot 
lie  fulcTuni 

rr   CM   =,«; 
!<!     0  =    tlu' 

when  tlicrc 


ex 


Sill 

6»  =  0; 

(1) 

•iiim 

.    Tlic 

It'll 

/'  iiiiil 

Ull. 

s   tl 

lit  .  I'oi' 

ciini 

lu  Ihr 

IV     1 

II'   M'll- 

11    \i 

llllr   III' 

n.sibilit}-  i.s 

i{h:qfis]Ti:s  or  a  aooD  balasce. 


187 


greater  the  sniuller  the  value  of  /'  —  W  is  ;  hence  tlie  sen- 
sibility may  bemeasuied  l.y  y,'^  ",|"  "^^^'i^'^  requires  that 

be  as  small  as  possible.  TlierefoR'  a  must  be  large,  and  w, 
li,  and  k  must  i)e  small  ;  /.  c,  the  arms  must  be  long,  the 
beam  ligiit,  and  tiie  distances  of  the  fulcrum  from  the 
beam  and  from  the  centre  of  gravity  of  the  beam  must  be 

small. 

The  llnril  requisite,  its  stability,  is  greater  the  greater 
the  moment  oi!  the  forces  wiiich  teiul  to  restori'  the  beam  to 
Its  former  i)osition  of  rest  when  it  is  disturbed.     If  P=  W 

this  moment  is 

[(y  +  II  )/,  +  wk]  sine, 

which  should  be  made  as  large  as  possible  to  secure  the 
third  requisite. 

This  condition  is,  to  some  extent,  at  variance  with  the 
second  requisite.  They  may  both  be  satisfied,  however,  by 
making  (/'  -f  M)  /'  +  n-k  large  and  a  large  also  ;  /.  v.,  liy 
iiicrearing  the  distances  of  the  fulcrum  from  the  beam  and 
from  the'c'cntre  of  gravity  of  the  beam,  and  by  lengthening 
(he  arms.  (Sec  Todhunter's  Statics,  p.  180,  also  Pratt's 
Mechanics,  p.  ^8.) 

The  coiiijiarativc  inii)orta'nce  of  these  qualities  of  w'».s'/- 
hililil  and  sluhiUhj  in  a  balance  will  depend  upon  the  use 
for  which  it  is  'intended  ;  for  weighing  heavy  weiglits, 
slithilihi  is  of  more  importance;  for  use  in  a  chemical 
laix.ratory  the  l)alanc.'  must  possess  great  snisihilily  ;  and 
instruments  have  been  constructed  which  indicate  a  varia 
tion  of  weight  less  than  a  miUhulh  part  of  the  whole.  In 
.,  halaiicc  of  great  di'licacy  tli.'  lulcrum  is  ma«le  as  thin  as 
possible;  it  is  gmcrally  a  h-iiifi-rilijr  of  liardened  sle.'l  or 
airate.  resting  on  a  polislicd  agi.tc  plate,  which  is  supported 
on  a  strong  vertical  iiillar  of  lirass. 


188 


rilE   SThKLVAJU). 


^ 


^ 


113.  The  Steelyard.— This  is  a  kind  <if  l)iil;uur  ii, 
whicli  \hv  anus  aw  iiiicmiiil  in  |(.|i<;tlK  (lie  loiiovr  one  hciu- 
f,n-ii(liiul(Hl,  along  wliicli  a  ;w/w  mav  he  moved  m  (.rd.r  u, 
liaiiUKv  (litl'i'ivnt  ucinlits  wliicii  aiv  plac'cd  in  a  scak'-pan  on 
tlio  short-arm.  Whilo  the  moment  of  the  substance 
vveiglied  is  elianged  liy  increasing  or  diminishing  its'(|nan- 
tity,  its  arm  remaining  eonslanl.  that  of  the  poise  is 
changed  l»y  altering  its  arm,  tlie  weiglit  of  tlio  poise 
remaining  the  same. 

114.  To   Graduate   the   Common   Steelyard.— (1) 

)Mn'n  the  pulnt  of  moipriisiun   is  (vnicli/v/d  ivilli   the  rciilrr 
of  (iravUy. 

I  vet  AF  l)e  tlie  beam  of  the  steel- 
yard suspended  about  an  axis  pass- 
ing through  its  centre  of  gravity. 
C  ;  on  the  arm,  ('!•",  jdace  !i  mov- 
ablo  weight,  /* :  (hen  if  a  weiglit. 
ir,  e<|ual  to  /',  is  suspended  from 
A,  (he  beam  will  balance  when  /' 
on   the  long  arm   is  at   a  distance 

from  V  efpial  (o  AC.  If  II'  e(|iials  twice  the  weigh!  of  /', 
the  beam  will  bahinee  when  (he  distance  of  /*  from  C  is 
twice  AO;  and  so  on  in  any  pr(>|(orl  ion.  Hence  if  IT  is 
successively  1  lb.,  'i  lbs.,  .'i  lbs.,  etc..  the  distances  of  the 
notches,  i,  "I,  ;{,  {.  etc.,  where  /'  is  placed,  are  as  1,  •>,  ;;. 
etc.,  /.  I'.,  the  arm  VV  is  divided  inio  ry//r// divisions,  bcLriu- 
ning  at  the  fulcrum,  ('.  as  the  zero  point. 

{•.')    Whi'u  Ihi'  poinl  of  sHsjinisioii  is  not  roi.in't/cn/  nilli 
flic  rent  re  of  i/niri/i/. 

Let  (' be  the  fulcriiin.    II'   the  siibstaiuv   to  be   wii-lied. 
banging  at    (he  e\(ivmi(y.  A,  and    /'  the  movable   wei-li! 
Suppose  that  when    11' is  ivmowd.  the  weight.   /'.  placd   at 
H  will  balance  the  long  arm.  ('I',  ami  keep  Ihi'  slce!\ard    in 
u  horizontal  position;  (hen  the  inoinent  of  (he  insdunieiit 


((C)      (s)|-UJ4_ij_Lf]xixxdl 

Fig.58 


f  biil.'iiici'  ill 
er  oiu'  hciiii^ 
!  Ill  (irdiT  1(1 

SC'llk'-|i;lll  (Ml 
0      Slll)St;|l|(H' 

i.ir  its'{iuiiii- 
Ik'  ])()isi'  is 
if    tlio  poist' 


lyard.— (1) 

h   the  ceiilrr 


LUi-Lftl 
0     -^ 


rTT-rrm-n-T-ril 
P 


ig.58 


fi.iiiit  of  /', 
'  froin  (!  is 
lice   ir   \V  i.s 

MCCS    of    (ill' 

lis  1,  -.>.  ;;. 
Foils.  Iicifiii- 

icidcnt  nilli 

i('   \vci"lic(|. 
Iilc    uciuli! 
'.  phic'il  Ml 
Iccly.ir.l    ill 
iiislruiiit.'iil 


KXAMI'IjKS. 


189 


ilsi'lf,  ;il)oii(  C,  is  on  tlio  '  k'.  <'l''.  iiiul  is  ('i|U;il  to  /'•('!>. 
llriicT,  if  M  luui;j;s  from  A,  ami  I'  from  luiy  iioiiit,  K,  llici' 
for  rciuilibrium  \\v  iiiu>t  iuive 

P.CE  +  r-  lU!  =  11'.  AC; 

or  p.  BE  =  ir.  AC; 


BE  =    ,•  AC. 


If  we  niiik(!  II'  succpssivoly  ('(|iiiil  to  /'.  ^>/',  HP,  etc..  tlion 
llu'  viihu's  of  BK  will  he  AC,  '.'AC.  :5A('.  etc.,  ami  tlH'so 
ilisiaiii'i's  must  he  mcasiiivd  oil',  comiiiciiciiiir  at,  H  for  tliu 
■/.wo  jjoiiil,  ami  tiii'  points  so  (k'termiiied  marknl  1,  i,  ;},  4, 
etc.  Such  a  steelyard  eaiiuot  \veij,di  lielow  a  eertaiii  limit, 
(•orrt'spoiiding  to  the  lirst  noteli,  1. 

To  lind  tl)"  length  of  the  divisions  on  the  beam,  divide 
i?K.  the  distance  of  the  poise  from  the  zero  point,  hy  the 
weight,  ir,  which  /'  balances  wiien  at  tiie  point  E.  The 
steelyard  often  has  /iri)  fnlcrnms,  one  for  small  and  the 
other  for  lartfe  weights. 


EXAMPLES. 

1.  What  force  mnst  be  api)lied  at  one  end  of  a  lever 
12  ins.  long  to  raise  a  weight  of  '.'>()  ll)s.  iianging  4  ins.  from 
the  fiilcrnm  which  is  at  the  other  end,  and  what  is  the 
pressure  on  the  fulcrum  ?  A)is.   lU  lbs.  :  20  lbs. 

2.  A  lever  weiglis  3  lbs.,  and  its  weight  acts  at  its  middle 
point  ;  tlie  ratio  of  its  arms  is  1  :  '.].  If  a  weight  of  4S  lbs. 
be  hung  from  the  end  of  the  shorter  arm,  what  weight 
must  l)e  suspended  from  tln'  other  eiul  to  present  motion? 

J«,s.    l.J  lbs. 


190 


WHEEL  A.\D  AXLE. 


3.  Tlic  ariiis  of  a  ftni/  li'vor  are  :]  ft.  and  5  ft.  and  inclined 
lo  eat'li  oilier  at  an  an<j;le  0  =  {'){f.  'J'o  the  .sliort  arm  a 
weijilit  of  T  ll)s.  is  applied  and  to  the  long  arm  a  weight  of 
(!  li)s.  is  apiijieii.  Re(jiiiri'd  (lie  inelination  of  each  arm  to 
the  horizon  when  there  is  erinililirinm. 

Alls.   The  short   arm   is  inolinod  at  an  angle  of  18°  2'i' 
(tboi'c  the  horizon,  and  the  long  arm  i.s  inclined  at  an  angle 
of  48'  'i'i'  hchiir  the  horizon. 


115.  The  Wheel  and  Axle.— This 
machine  con-ists  ol'  a  wheel,  a,  rigidly 
connected  with  a  horizontal  cylinder, 
//,  movable  ronntl  two  trunnions  (Art. 
!)!)),  one  of  which  is  shown  ;  t  r.  The 
power.  /',  is  applied  at  the  circnnifer- 
eiico  of  the  wheel,  sometimes  by  a  cord 
coiled  round  the  wheel,  sometimes  by 
handspikes  as  in  the  ciijistini,  or  by 
handles  as  in  the  iriiii/ldss  :  the  weight,  If,  hangs  at  the 
end  of  a  cord  fastened  to  the  axle  and  coiled  rouiid  it. 


116.  Conditions  of  Equilibrium  of  the  Wheel  and 
Axle.  (1)  Let  a  and  0  be  the  radii  of  the  wheel  and  a.\le 
rt'spectively  ;  /'and  H'  the  power  and  weight,  sujiiiosed  to 
acl  by  strings  i't  tlie  <  innmference  of  the  wheel  and  axle 
perpendicular  to  tlie  radii  a  and  /j.  Then  either  i)y  the 
principle  of  virtual  velocities  or  l)y  the  princiide  of  momenta 
wc  have 


or 


Pa  =  ]\1>, 

P  radius  of  axlo 

W  ~  railing  of  wheel 


0) 


Tt  is  evident  thai,  by  increasing  the  radius  of  (he  wheel 
(ir  by  diminishing  (he  I'adiiis  of  the  axle,  any  amount  of 
nicchaiiieal  advantage  may  be  gained.     It  will  also  be  seen 


1(1  inclined 
lurt  arm  a 
I  weight  of 
ich  arm  hi 

of  is°  -z-r 

it  an  an<rk' 


ngs  at  the 
iid  it. 


Theel  and 

1  and  axle 
iil)[)osed  to 
L'l  and  axle 
her  l)v  the 
if  momenta 


0) 

f  the  wIhtI 
amount  of 
l.so  l)e  seen 


niFFKiiEyriAL  wiiehl  a.\d  axle. 


191 


that  this  machine  is  only  a  modification  of  the  lever  ;  the 
peeiiiiar  advantage  of  tlic  wheel  and  axle  being  that  an  end- 
less series  of  levers  are  brougiit  itito  jilay.  In  tliis  res|K'ct, 
tlion,  it  surpasses  the  common  lever  in  mechanical  advan- 
tage. 

In  the  above  we  have  sii|i[)osed  friction  to  be  neglected, 
or,  wliat  amounts  to  the  same  thing,  iiave  assumed  that  the 
trnnnion  is  indednitclj  small.  In  practice,  of  course,  the 
trunnion  has  a  certain  radius,  r.  and  a  certain  coefHcicnt  of 
friction.  Cilling  A' tlie  resultant  of  /'and  H',  ami  takiug 
into  account  the  friction  on  the  trunnion  we  have  for  the 
relation  between  /'  and   W 

Pa  =   Wl)  -j.  ,•  ,siu  (ji  \//'2T  ir^~+^^/'lT77)S  (.),      i^i) 

M  being  the  angle  between  the  directions  of  P  and  U' 
exactly  as  in  Art.  110. 

(v')  Differential  Wheel  and 
Axle. — By  diniinishing  li,  the  radius 
of  the  axle,  the  strength  of  the 
machine  is  diminished  ;  to  avoid  this 
disadvantage  a  (liJfi'nuiKil  wheel  and 
axle  is  sometimes  employed.  In  this 
instrument  liie  axle  consists  of  two 
cylinders  of  radii  h  and  It  ;  the  rope 
is  wound  round  the  former  in  one 
direction,  and  a^ter  jiassing  under  a 
movable  pulley  to  which  the  weight 
is  attached,  is  wound  niuiul  the  latter  in  the  opposite  direc- 
tion, so  that  as  the  p.iwer,  /',  which  is  applied  as  befiu'c, 
tang<'iitiMlly  to  the  wbeel  of  radius,  a.  moves  in  its  own 
dir.'ctioii,  I  he  rope  at  b  winds  up  while  the  rope  al  // 
unwinds. 

l''(M' the  ei|uililiriuui  of  the  forces  (whether  at  rest  or  in 
uniform  motion),  the  tensions  of  the  rope  in  bm  and  b' n 


ia2 


TOOTH/. I)    WIlhKLS. 


art"  each  equul  to  iW.     lli'iui',  taking  moiiu'iiis  round  tlm 
ceiiLrc  of  the  truniiiuii,  c,  we  havi' 


Pa  +  iWb'  -  illV;  =  0; 
•.    Fa  =  l\l{b-  l>'), 


(3) 


hence  l)y  makiiij]?  tlie  iliiTerenec,  b  —  b',  small,  the  power 
can  1)0  nuide  as  .<mall  as  wo  pleuso  to  lift  a  given  weight. 
Let  tlie  wheel  turn  throngh  tiie  angle  (SO;  the  point  of 
apjilication  of  /'  will  descrilie  a  space  -=  «(5W,  and  the 
weiglit  will  lie  lifted  tlirough  a  space  =  ^  {b  —  b)  '50, 
which  lattoi-  will  he  very  small  if  b  —  b'  is  very  small. 
Therefore,  since  the  amount  of  work  to  he  done  to  raise  t\\v 
weiirht  to  anv  gisen  heigiit,  is  constant,  economy  of  i)ower 
is  accomplished  hy  a  loss  in  tiio  time  of  performing  the 
work. 

117.  Toothed  Wheels.— 7W/<rr/  or  mfned  vhcvh  are 
wheels  ]»rovided  on  tlie  circumt'eronces  with  projections 
called  teetli  or  cogs  which  interlock,  as  shown  in  the  figure, 
and  which  are  therefore  capaiile  of  transmitting  foi'ce,  so 
that  il"  one  of  the  wheels  he  turned  round  hy  any  means, 
the  oilier  will  be  turned  round  also. 

Wiien  the  teeth  are  on  liic  sidcx  of  the  wheel  instead  of 
the  circuml'erence.  they  are  called  crowa  wiieels.  When 
tiie  axes  of  two  wheels  are 
neither  perpendicular  nor 
parallel  to  each  other,  tlio 
wheels  take  the  form  of 
fru-tums  of  cone^.  and  are 
called  hi  rrlrd  wlicrlx.  When 
tlicii'  is  a  i)air  of  toolhcd 
v\  III I'ls  on  each  axle  with  the 
teeth  of  the  laruv  one  on  one 


axle  titting  between  the  teeth 


Fig.61 


US  round  tlia 


(3) 

ill,  the  power 
a;iveii  weiglit. 
the  point  of 
kV),    iind    the 

\  [b  -  b)  r5«, 

s  very  small. 
le  to  raise  the 
)my  of  i)()\vi'r 
rlbrniing  the 


led  v'hcch  are 
1  project  ions 
in  the  tiuure, 
ting  force,  so 
ly  any  means, 

leel  instead  of 
heels.     When 

Q 


Fig.6l 


TOOTJIi:!)    WIIKKLS. 


193 


of  the  snuill  one  on  tlie  next  axle  tlie  larger  wlieel  of  each 
pair  is  called  tiie  wheel,  and  the  smaUer  is  called  the  yv/,/ w/. 
By  means  of  a  eonihination  of  toothed  wiieels  of  tliis  kind 
called  a  train  of  wheels,  motion  may  he  transtVrred  trom 
one  point  to  anotlicr  and  worl<.  tlone,  each  wheel  driving 
the  next  one  in  the  series.  The  discussion  of  this  kiml  of 
niaciiinerv  possesses  great  giv)metric  elegance  ;  hut  it  would 
he  (Hit  of"  place  in  this  work.  We  shall  give  only  a  sliglit 
sketcli  of  tiie  simplest  case,  that  in  which  the  axes  of  tlie 
wheels  are  all  i.arallel.  For  the  investigation  of  the  proi)er 
forms  of  teeth  in  order  tliat  the  wheels  when  made  shall 
rnn  trnly  one  upon  another  the  student  is  referred  to  other 
works.* 

118.  To  Find  the  Relation  of  the  Power  and 
Weight  in  Toothed  Wheels.— 1-ct  A  and  U  he  the  lixcd 
centres  of  the  toothed  wlieels  on  tlie  ciuHimfercnces  of 
whicli  tlie  ^?eth  are  arranged  ;  QCQ  a  normal  to  tlie  sur- 
faces of  two  Veetti  at  their  piuiil  of  contact.  ('.  Suppose  an 
axle  is  fixed  on  llie  wheel.  H.  and  llie  weigiit,  W,  suspended 
from  it  at  K  iiy  a  cord  :  also,  suppose  the  jiower.  /'.  acts  at 
1)  with  an  arm  DA  ;  draw  Xa  and  15/  [U'l-pendiciilar  to 
QCQ.  Let  Q  he  tiie  mutual  pressure  of  one  tootii  upon 
another  at  C;  tliis  jtressure  will  he  in  tlie  directi.m  of  the 
normal  QCQ.  Now  since  the  wheel,  A.  is  in  eipiilihrium 
ahont  the  fixed  axis,  A,  under  the  action  of  the  forces, 
r  and  Q,  we  have 

(1) 


P-  AD  -  Q-  Art; 


and  since  the  wheel,   B,  is  in  e.pnlihriiim  ahont  the  fixed 
axis,  B.  under  the  action  of  tlie  forces.  Q  and   If,  we  have 


11'.  HK  =r  n.m. 


i-i) 


*  Soe  GoodoveV  m^uml^of  ifeeha'iism:  Huiikiro's  .l/i/ilh,!  M.cha>,U'x  :  Mmsc 
l,..v'H  Knqlmering;  WiUiH'H  Pnnci,>le,<  of  .\fec/,uni.-m  ;  (;(illi-iioir.>^  SlaHqtie :  and 
a  Pajyer  of  Mr.  Airy's  in  Ike  Camb.  PhU.  Trans..  Vol.  II,  p.  3TI. 
5) 


1!»4 


TI:aI.\    (,F    \     WHEELS. 


Dividinj?  (I)  by  (:*)  wc  luivo 


or 


If.  BE  ~  B6' 

moment  of  P Ka 

moment  of  IT  ~  Bd' 


If  the  direction  of  tiie  normal,  QC'Q.  at  the  point  of  con- 
taet,  C.  eliiiiiges  as  tlie  action  passes  froii  one  tooth  to  the 
snceei'dinij:,  tlie  rehition  of /'  to  IT  becimies  varial)le.  But. 
if  the  teeth  are  of  siieli  f(jrm  that  tlie  normal  at  their  point 
of  contact  shall  always  he  taniroiit  to  hoth  wheels,  the  lines 
An  and  BA  will  l)ecome  radii,  and  their  ratio  constant. 
And  since  the  nnmher  of  teeth  in  the  two  wheels  is  propor- 
tional to  their  radii,  we  have 


moment  of  P  _  niind)er  of  teeth  on  the  wheel  /' 
•moment  of  IF  ~  number  of  teetli  on  the  wheel  ()"' 


(=5) 


119.  Relation  of  Power  to  Weight  in  a  Train  of  n 

Wheels.— Let  R^,  J?,,,  R^,  etc.,  be  the  radii  of  the  suc- 
cessive wheels  in  such  a  train  ;  r,,  r^,  r^,  etc.,  tl>e  radii  of 
I  he  corresponding  pinions;  and  let  P,  P,,  Pj,  P3,  .  .  .  \y, 
be  the  powers  ap{)licd  to  the  circumferences  of  the  successive 
wheels  and  i)inions.  Then  the  first  wheel  is  in  eciuilibrium 
about  its  axis  under  the  action  of  the  forces  P  and  /',, 
since  the  power  applied  to  the  circumference  of  t!ie  second 
wheel  is  equal  to  tiie  reaction  on  the  tirst  pinion,  therefore 


Similarly 


P  X  h\ 

Pi      X     Po 


P,  X  /.\ 


etc. 


P,    X   Ti. 
2    '^   ^8  > 

etc.; 


-Pfi-l  X  H„  =  W  X  rn. 


EXAMPLES. 


1!»5 


I' 


point  of  con- 
tooth  to  tilt' 

ri!il)le.  But, 
t  thtir  point 
els,  the  lines 
tio  constant, 
ils  is  jiropor- 


hecl  P 
heel  I)'* 


(a) 


,  Train  of  n 

of  the  snt'- 
the  radii  of 
P„  .  .  .  W, 
he  siirecpsive 
equilihrinni 
I  Piind  /',, 
f  tlie  second 
1,  therefore 


Miiliililvin^r  these  eciuations  together  and  omitting eouimon 
factors,  we  have 


P 


r,  X    r.,  X    r 


X 


A'l  X  Rz  X  /.'s 


(1) 


It  will  be  observed,  in  toothed  gearing,  that  the  smaller 
the  radius  of  the  pinion  as  ccmipiired  with  the  wheel,  tin- 
greater  will  be  the  mechanical  ad\antage.  There  is,  how- 
ever, a  [iractical  limit  to  the  size  that  can  be  given  to  the 
pinion,  because  the  teeth  must  be  large  enough  for  strength. 
;iiid  must  not  be  too  few  in  number.  Six  is  generalh  the 
lea^t  number  admissible  for  the  teeth  of  a  pinion.  E(iua- 
tiun  (1)  shows  that  by  a  tram  consisting  of  a  very  few  pairs 
of  wlieels  and  pinions  there  is  an  enormous  nieehanical 
advantage.  Thus,  if  there  arc  three  pairs,  and  the  ratio  of 
each  wheel  to  the  pinion  is  10  to  1,  then  P  is  mily  mie 
tiiousandth  part  of  U';  but  on  the  other  liiind,  IT  will  only 
make  one  turn  where  /'  makes  one  tiiousnnd.  Such  trains 
of  wheels  are  very  useful  in  machinery  such  as  hand  cranes, 
where  it  is  not  essential  to  obtain  a  nuick  motion,  and 
where  the  power  available  is  very  small  in  comparison  to 
the  weight.     (See  Browne's  Mechanics,  p.  lUU.) 

EXAMPLES. 

1.  What  is  the  diameter  of  awheel  if  a  jxiwer  of  :$  lbs. 
is  just  able  to  move  a  weight  of  l".'  lbs.  that  hangs  from  the 
axle,  the  radius  of  the  axle  l)eing  'l  ins.?        Ans.   IC  ins. 

'i.  If  a  weight  of  ;iO  lbs.  be  s!ii)ported  on  a  wheel  and 
axle  by  a  f>rce  of  4  lbs.,  and  the  radius  of  the  axle  is 
5  in.,  find  the  radius  of  the  wheel.  -!«•'>■•   -'i  i'-"'- 

;*).  A  capstan  is  worked  by  a  mim  lushing  at  the  end  of 
a  pole.  lie  exerts  a  force  of  .")(>  liis.,  and  walks  lo  fl. 
roiiml  for  every  'i  ft.  of  rope  pulled  in.  What  is  the 
resistance  overcome  J*  -^l"*-  ^y"  1''3. 


196 


l.\C/J.\hl>    PLASE. 


4.  An  axlo  wliosc  dianalir  is  lU  ins.,  has  on  it  two 
wlu'C'ls  the  (liaiiietcrs  of  wliich  aiv  'i  If.  and  vlj  ft.  rcspoo 
tivi'ly.  Kiiitl  the  wt'jolit  tlial  would  lie  sujipoitcd  on  llic 
axli;  liy  \Vfiiri:ls  of  :*.")  Il)s.  and  ;IA  Itis.  on  iIil-  sinalh'i'  and 
hirgcr  wliofis  ivspoctivok.  Anx.    V.Vl  il»s. 

120.  The  Inclined  Plane  -Tliis  lias  alivaiiy  lurn 
partly  considtTcd  (.Vrt.  '.Mi.  clc).  Let  the  power.  /'.  wliose 
dircclion  makes  an  an'j;le,  0.  with  a  ronirli  inclined  plane, 
lie  employed  to  drai;  a  weiirht.  IT,  np  the  ]ilane.  Then  if 
0  is  the  aiiirle  of  friction  and  /  the  inclination  ot"  tlie  piano, 
we  have  from  (3)  of  Art,  !)(], 


P=   II 


.sin  {i  +  (/.) 


cos  (r^  —  0) 

If  /'  acts  along  the  jilaiie,  0  =.  0,  and  (1)  hocomes 

.sin  (/  +  0) 


(1) 


p=  ir 


cos  f/i 


(^) 


If  P  acts  horiz(>ntaily,  0  =  —  /,  and  (I)  hecoraes 

I'  =   It  tan  (t  +  ^).  (3) 

0()i!.-^lf  WO  suj)j)oso  the  friction  =  0,  (1),  (2),  and   (3) 
hocomc  resi)octivoly 

,sin  i 


ir 


cos  Q' 


(4) 


7'=  II  sin  /,  (5) 

/•=   W  Ian  /.  (d) 

ScH.— It  follows  from  (I),  (5).  and  (C)  that  the  smaller 


its  (III    it    two 

v'i   I't.  ri'SpLT- 

iiitcd  on   ilif 
.sMiiillcr  iuid 
IS.    \:]-l  Ihs. 

iilrcady  lircii 
*VlT.  /'.  whoso 
fliiu'il  ])Iaiic'. 
lie.  Thoii  if 
of  tlie  piano, 


(1) 


.'conies 


comes 

(3) 
{2),  and  (;5) 

(4) 

till!  .sinallor 


77/ A-  rii.t.nr. 


19? 


tile  inclinution*  of  tlic  |ihinc  lo  lin'  lur i/on.  the  j/rcatiT  will 
lie  tlif  inrcliaiiical  advanlam'.      If  uc   take  in  friction  tiitic 

i>    an    ('\cc|M  ion    to     iliis    rule     when     /  >  _^  _  (/,.       'I'li,. 

gradii'nts  on  railways  arc  tlio  ni.ist  comnion  cxainiilcs  ol' 
till'  iisf  of  llio  incluK'd  plant';  |Ik'?;c  arc  always  niadi'  a>  |nw 
a.s  IS  fonvi'niont  in  orik'i'  tu  inabk'  tin'  cnyint,-  to  iili  tiu' 
lieaviest  possii)k'  train. 

121.  The  Pulley. — Tlie  pulfn/  oonsisfs  of  a  i/ninrri/ 
?rArr/,  capaljlo  of  iL'Voh ing  freely  about  an  axis,  lixcd  into 
u  framework,  called  the  block.  A  cord  passes  over  a  por- 
tion of  the  circinnfererue  of  tlie  wlictd  iji  the  groove. 
When  the  axis  of  tiie  i>uliey  is  tixed.  tlie  pulley  is  called  a 
lij-i'd  pulley,  and  its  only  effect  is  to  c  '  ange  the  direction 
of  the  force  exerteil  liy  tiie  cord  :  hut  where  the  pnlley  can 
ascend  anil  descend  it  is  called  a  innrdblc  pnlley,  and  a 
lueclianical  advantage  may  lie  gained.  Condiinations  of 
pulleys  may  !)e  made  in  endless  variety  ;  we  shall  consider 
only  the  simple  nujvaiile  i)ullev  and  three  of  the  more 
ordinary  eomhinations.  No  account  will  lie  here  taken  of 
the  weight  of  the  pulleys  or  of  the  cord,  or  of  friction  and 
stiffness  of  cords.  The  weight  of  a  -et  of  pulleys  is  gener- 
ally small  in  comparison  with  the  loads  which  they  lift: 
and  the  friction  is  small.  Tlie  use  of  the  pulley  is  to 
diminish  the  elfects  of  friction  which  it  does  by  transferring 
the  friction  between  the  cord  ami  circumference  of  the 
wheel  to  tlio  axis  and  its  supports,  which  may  be  highly 
])olislied  or  lubricated.  The  mechanical  principle  iinohed 
in  all  calculations  with  respect  to  the  pulley  is- the  constancy 
of  tiie  force  of  tension  in  all  parts  of  the  same  string 
(Art.   to). 

*  To  find  llic  inrlinaiion  of  llu-  rl'""'  f'""  n  iii.Txiiiiiim  viiliic  of  P  win  u  it  aits 
pariilli'l   to  tlie  ptaiio  wo  put  the  ili'i-ivMtiM'  cf  /'  wiili   rp-^jn'it   to  i      il.  and  t;rt 

j  ,/i      Iti'ii  '■  wl.i'o  llii'  iiiiliiiatlon  of  tln'  plane 

i/.,  inccliaiiical  advuiitat;.'  i-  ■liiiiiiii-^liiiit% 


Hi 


W 


•p) 


cos  0 


ih  dluiiiii.-'liiiii'  from  ,  to 
2       2 


11)8 


FII{ST  SYSTKM    OF   ITLLEYS. 


122.  The  Simple  Movable  Pulley.— Lot  0  bo  Ww 

(•('iilic  of  the  ])iillev  wliicli  is  siiiij)ortt'(l  by  a  cord  passiiit: 
iiiidiT  it,  «i!li  oil"  ciiil  attiiciiod  to  a  licaiii  al  A  and  tlu' 
oIIkt  end  .sti't'tciicd  by  tbc  force  /'. 

Now  since  tiic  tension  of  the  string, 
A  1)1)1',  is  tiiosanu;  tlirt)uglioiit,  and  tlie 
Wright,  ir,  is  supported  by  the  two 
strings  at  IJ  and  D,  in  eueli  of  which 
the  tension  is  P,  we  have 


2P  =  W; 


P 


1 


Fig.62 


Tlic  same  result  follows   by  the  prin- 
ciple of  virtual  velocities.      '.appose  the 
l)ulley  and  the  weight,  H',  to  rise  any 
distance.      Tlien   it  is  clear  that  both  halves  of  the  string 
must   1)0   shortened    by   the   same  distance,  and  hence   /' 
must  rise  double  the  distance ;  and  therefore  the  ctpiatiou 
of  virtual  work  gives 

P       1 

3/'  =  W;     .  ^ 


IV 


3 


The  mochanieal  advantage  with 
is  2. 


123.  First  System  of  Pulleys,  in  which 
the  same  cord  passes  round  all  the  Pul- 
leys.— In  this  system  there  are  two  blocks,  A 
and  B,  the  upper  of  which  is  lixed  and  the 
lower  movai)le,  and  each  containing  a  number 
of  pulleys,  each  pulley  being  movable  round 
the  axis  of  the  block  in  wliicli  it  if.  A  single 
cord  is  attached  to  the  lower  block  and  pa.s.ses 
alternately  round  the  pulleys  iu  the  upper  and 
lower  blocks,  the  portions  of  I  lie  cord  between 
successive  pulleys  being  parallel.    The  portion 


a  single  movable  pulley 


.ot  0  bo  tlu' 

cord  pass  lilt: 

ill   A  iiiiii  till' 


Fig.62 

of  llu'  striiif^ 
mil  luiire  /' 
tlie  c'i[iiiitioij 


Fifl.63 


finST  SYSTEM    OF  VrLI.FAS. 


195t 


of  cord  prococdiug  from  one  pulley  to  the  next  is  ealloil  ii 
ply;  Uk'  portion  at  which  the  powi'r,  /'.  is  iippHtd  is 
called  the  hirk-le-faU. 

Since  the  cord  jiasses  round  all  the  pulleys  its  tension  is 
the  same  throuaiiout  and  eipial  to  /'.  'I'lieii  if  n  l)e  the 
nuniher  oi'  jilies  at  the  lower  block,  nP  will  lie  the  resultant 
upwaicl  tension   of  the  cords  at  the  lower  block,  which 

must  equal  \\  ; 

.-.    nP  =  W, 


or 


P 

W 


1 

n 


This  result  follows  also  liy  the  principle  of  virtual  vcloci- 
ties.  Let^  denote  the  length  of  the  tackle-fall  and  x  the 
common  length  of  the  jdies  ;  then  since  the  length  of  the 
cord  is  constant,  we  have 

p  4-  n.r.  =  constant ; 

.•.   dp  4-  ')ulx  =  0. 

But  the  equation  of  virtual  work  is 

Pdp  +  Wdx  =  0; 

IF  PI 


P  = 


n 


or 


W 


n 


This  system  is  most  commonly  ust.l  on  ncconnt  of  its 
superior  portaliility  and  is  the  only  one  of  practical  impor- 
tance. The  several  jiulleys  are  usually  mounted  on  a  com- 
mon axis,  as  in  the  figure,  the  cord  being  inclined  slightly 
(imde  to  jiass  from  one  jniir  of  pulleys  to  the  next. 

This  forms  what  is  called  a  set  of  7iJnr/,-s  and  Fnl/tt.  It, 
is  very  commonly  used  on  shipboard  and  wherever  weights 
have  to  l»e  lifted  at  irregiiliir  times  and  places.  'l"he  weight 
of  the  lower  set  of  pulleys  in  this  case  merely  forms  part  of 
tlu'  gross  weight  \V. 


'^'00 


.VAvo.v/;  .s).sy/-;.v  of  I'ILI.kys. 


Tlu'  i'ri('ti<iti  <iti  tlic  siiiiullo  of  ;iiiy  partifuliir  p-.illcv  is 
lirii|)()rli(in;il  to  llic  total  pressure  on  (lie  jmlley,  wliicli  is 
{•k'iirlv  •.'/'.  Ilfiico,  if  n  is  llic  footliciciil  of  iriction.  tlu' 
rcsisliiiicc  of  Iriction  (in  any  imlliy  =  :.'/'//;  ;inil  ilic 
unioiini  of  its  (lis|)la(cnirni.  wlicn  IT  is  raised,  will  be  to 
the  diT^plaeemont  oT  11  in  tlie  ratio  of  tiic  imliiis  of  the 
s](iiidle  to  tluifc  of  Ihc  jjiiiley. 


X 


124.  Second  System  of  Pulleys,     [ 
in  which  each  Pulley  hangs  from  a 
fixed  block  by  a  separate  String. —      '^ ' 

Let  A  lie  the  lixi-d  jmlley,  n  the  number        p 
of  movable  ]uilleys  ;  each  cord  has  one 
end  aitarhed  lo  atixrd  point  in  the  beam, 
and  all  exeejit  the  last  liavr  tiie  oIIut  end  Qw 

attached   to  a   movalije    i   illey,   the  jior-  Fig. 64 

tions  not  in  eontaet  witii  any  pniley  iieini,'  all  parallel. 
[I'heii    the   tension    of   the   cord    passing   under    the   lirst 

(lowest)  pulley  =    _^   (Art.  1'.'^)  ;  the   tension   of  the  eord 

Ijassinif  under  the  second  pulley  —    ^j,  and  so  on  ;  and  the 

tension  of  the  cord  passiiii^  under  the  «th  pulley  ^  -- , 
wiiieh  must  eipud  tlie  power,  1' ; 


/'  _  J. 

W  ~  2«' 


0) 


The  same  result  follows  by  the  prineinle  of  work.  Sup- 
pose tile  lirst  pulle\  and  the  weiirlit  IC  to  rise  any  di.^taiiee. 
:r  \  then  it  is  clear  that  lioth  portions  of  the  cord  pa.ssinjj 
round  this  pulley  will  be  shortened  liy  the  saiii(>  distance, 
and  hence  the  second  pulley  must  rise  double  this  distaiiee 
or  "ix,  and  the  third  pulley  must  rise  double  the  distance  of 
the  second  or  5i'.''.  and  so  on  :  and  tlie  )i\\\  ])iilley  niuKt  rise 
!i" "■'.<■  and    /'must  descend    ;.'".(:  llierefore  the  work  of  /' 


uliir  jvillcy  is 
lley,  wliicli  is 
i'l'ictidii.  the 
'/'  ;  iiiid  I  lie 
'd,  will  1)1'  to 
l-adiiis  oi'  the 


Fig. 64 

|)iiriill('l. 

(Icr   1li(^  lirst 

(if  the  cold 

(in  ;  1111(1  tJK! 
11  "' 


0) 

work.  Suit- 
any  (li.-llllH'C. 

cord  passinjj 
iiii(>  dislancc, 
tliis  dislaiu'o 
ic  distance  of 
ley  niiist  rise 
le  work  of  /' 


TIIIHI)  SVSTKM  OF  PVH.EYS. 


201 


is  P-i'^.r,  and  tlio  work  to  lie  done  on  IT  is  W-t.     llonce 
the  eciuation  of  work  gives 

P        1 


V.-VKc  =  \Vx, 


'    w 


125.  Third  System  of  Pulleys,  in  which  each  cord 
is  attached  to  the  weight.— In  this  system  one  end  of 
each  curd  is  utLached  to  the  bar  from  which  the  weight 
hangs,  and  the  other  supports  a  pulley,  the  cords  being  all 
parallel,  and  the  number  of  nioval)le  pulleys  one  less  than 
the  numlier  of  cords. 

Let  n  be  the  number  of  cords;  then  the 
tension  of  the  cord  to  which  /'  is  attached  is 
P ;  the  tension  of  the  senmd  cord  is  2/'  (Art. 
\n)',  that  of  the  next  -^-i/',  and  so  on;  and 
tli(!  tension  of  the  «th  cord  is  3«-iF.  Then 
the  sum  of  all  the  tensions  of  the  cords 
attached  to  the  weight  must  e(iual  W. 
Hence 


P  4-  2i'  +  2«P  +  . 


Fig:65 

.  2"-!/'  =  (2'«  -  1)  /»  =  W) 


W 


2«-  1 


In  this  system  the  weights  of  the  movable  jiuUeys  assist  P\ 
ill  the  two  former  systems  they  act  against  it. 

EXAMPLES. 

1.  What  force  is  necessary  to  raise  a  weight  of  480  lbs. 
by  an  arrangement  of  six  iiulleys  in  which  the  same  string 
passes  round  each  pulley  ?  Ans.  80  lbs. 

2.  Find  the  power  which  will  8U])port  a  weight  of 
800  lbs.  with  three  movable  pulleys,  arranged  as  in  the 
sec(md  system.  Ans.  100  lbs. 


•^02 


rut:  WEDCK. 


.'5.  If  tluTc  1)1"  ('(|iiilil)iiiim  In'twot'ii  /'  iitid  W  witii  three 
|piilli'vs  in  tlic  thinl  -vstciii,  wlial  additional  wci.vlit  can  be 
raised  if  •>  U.S.  he  added  to  r>  Aii.^.   14  lbs. 

126.  The  Wedge. — Tiic  weilgp  is  a  triaiipnlar  prism, 
iisualiy  isosceles,  antl  ia  used  for  .separating  bodies  or  jjarts 
of  tlie  same  body  l)y  introducing  ilo  edge  l)et\veen  them  and 
tlien  tiinisiiiig  liie  weilge  forward.  This  i.s  effected  by  the 
iijow  of  a  hammer  or  other  such  means,  which  produces  u 
violent  ))re.ssnre,  for  a  short  time,  in  ii  direction  i)erj)en- 
dicular  to  tiie  back  of  tiie  wedge,  and  tlie  resistance  to  1h? 
overcome  consists  of  friction  and  a  reaction  due  to  the 
molecular  attractions  of  the  particles  of  the  l)ody  which 
are  Ix'ing  separated.  Tliis  reaction  will  be  in  a  direction 
perpendicular  to  the  inclined  surface  of  the  wedge. 

127.  The  Mechanical  Ad- 
vantage of  the  Wedge.— Let 
ACH  rejiresent  a  section  of  the 
wedge  peri)endicular  to  its  in- 
clined laces,  the  wedge  having 
been  driven  into  the  nuiterial  a 
distance  ecpial  to  DC  by  a  force, 
i-*,  acting  in  the  direction  DO. 
Draw  DE,  DF,  jierpe.ulicuhir  to 
AC.   1K;.   and   let    7.'  denote   the 

reactions  along  ED  and  FD  ;  then  fiR  will  bo  the  friction 
acting  at  K  and  Fin  the  directions  EA  and  FH.  Let  the 
angle  of  the  wedge  or  AVH  =z  2«. 

He.solve  the  forces  which  act  on  the  wedge  in  directions 
perpendicular  and  j)arallel  to  the  back  of  the  wedge,  iheu 
we  have  for  perpendicular  forces 


Fig.66 


P  =:  2/?  sill  rt  -f  2/tA'  cos  rt. 


(1) 


This  equation  may  aho  hr  ohfninod  from  the  principle  of 
work  as  follows:    If  the  we  ige  has  been  driven  into  the 


witii  Uuvv. 
''-'.Ill  fiiii  bo 
"■-.  i-ilbs. 

pillar  prism, 

i<'S  or  jiaris 

'(-'11  tliuiii  and 

ectod  by  tlio 

I  prodncf.s  a 
tion  j)crjic'n- 
staiinc  to  Im? 

due    to  tlio 
body  wliicli 

II  ii  direction 
dge. 


the  friction 
5.     Lvi  the 

n  directions 
«'(lgt',  Ihoii 


(1) 

^riuciph  of 
»i  into  Ihe 


i 

'I 


MKCllAMrM.   AhVA.\TAi! i:   til'   W i:ii(l !■:. 


2(i;i 


material  a  distance  ('(iiial  to  DC  by  a  force,  /',  acting  in 
tiic  direction  DC,  then  tiie  work  done  i)y  /'  is  /'  x  DC 
(Art.  101,  Weni.) ;  and  sinci'  tlie  pninis  E  aiul  F  were 
Mi'iginally  togcliier,  tbe  \vori\  done  against  tlie  resistance 
//  is  A'"x  DK  +  /.'  X  DF  =  27?  x  DK;  and  the  woriv 
done  against  friction  is  •.'/'/»*  X  KC.  Ilenec  tiie  e((iiatioii 
of  woriv  is 

P  X  DC  =  :>/;  x  DK  +  i,iR  x  EC,  (-J) 

wiiich  reduces  to  (1)  l)y  snl)stitntinir  sin  a  and  cos  «  for 
DE       ,  EC 
DC  ""^^  DC- 

Con. — If  friction  be  neglected.  {;i)  becomes 


r 
Ji 


'.'DE 
DC" 


AB 
AC' 


that  is 


P 

11 


back  of  the  wedge 


length  of  one  ot  the  ('(luul  sides 


It  follows  that  the  narrower  the  back  of  the  wedge,  the 
greater  will  lie  the  rncclianical  advantage.  Knives,  chisels, 
and  many  other  imidemcnts  are  exainjiles  of  the  wedge. 

In  the  action  of  the  wedge  a  great  part  of  tiie  power  is 
employed  in  cleaving  the  material  into  which  it  is  driven. 
The  forec  re((nired  to  effect  this  is  so  great  that  instead  of 
applying  a  eontinnons  pushing  force  jierpendiciilar  to  the 
back  of  the  wedge,  it  is  driven  by  a  series  of  blows.  Be- 
tween the  blows  tliere  is  a  jiowerful  reaction.  /?,  acting  to 
pn-li  the  wedge  back  again  out  of  tlie  cleft,  and  this  is 
resisted  by  the  friction  which  now  acts  in  the  directions 
VX  and  FC.  Hence  when  the  wedge  is  on  the  point  of 
starting  back,  between  the  blows,  the  eipnition  of  ecpii- 
librium  will  be  from  (1) 

Hit  sin  «  —  HfiR  cos  f e  =  0  ; 

.• .    «  =  tuir'  ^^. 


204 


THE  SrREW. 


And  tlio  wedgo  will  fly  buck  or  not  arcording  us  a  >  or 
<  tair^/t.  (See  Browne's  Mccliauics,  p.  117.  Also  Magnus's 
Mechanics,  p.  157.) 

128.  The  Screw. — The  screw  consi.>its  of  a  right  cir- 
cular cylinder,  on  the  convex  surface  of  which  tiiere  is 
traced  a  uniform  projecting  thread,  abed  ....  inclined  at 
u  constunt  angle  to  straight  lines  parallel  to  the  axis  of  the 
cylinder.  The  path  of  the  thread 
may  be  traced  by  the  edge  AC  of 
an  inclined  plane,  ABC,  wrajiped 
round  the  cylinder;  the  base  of 
the  plane  corresponding  with  the 
circnmferenco  of  the  cylinder,  and 
the  height  of  the  plane  witii  the 
distance  between  the  threads  which 
is  culled  the  pitch  of  the  screw. 
The  threads  may  be  rectangular  or 
triangular  in  section.  The  cylinder 
fits  into  a  block,  on  the  inner  sur- 
face of  which  is  cut  a  groove  which  is  the  exact  counterjiart 
of  t'.ie  thread.  The  block  in  which  the  groove  is  cut  is  often 
called  the  lud.  The  i)ower  is  generally  applied  at  the  end  of 
a  lever  fixed  to  the  centre  of  the  cylinder,  or  fixed  to  the  nut. 
It  is  evident  that  a  screw  never  requires  any  ])ressure  in  the 
direction  of  its  axis,  but  must  be  made  to  revolve  only  ; 
and  this  can  be  done  l)y  a  force  aiting  at  right  angles  to 
the  extremities  of  its  diameter,  or  its  diameter  produced. 

129.  The  Relation  between  the  Power  and  the 
Weight  in  the  Screw.— Su])pose  the  power,  P.  to  act  in 
a  plane  perpendicular  to  the  axis  of  the  cylinder  and  at  the 
end  of  an  arm,  DE  =  a,  and  suppose  the  screw  to  have 
made  one  revolution,  the  power,  /',  will  have  moved 
through  the  circumference  of  which  a,  is  the  radius,  and 
the  work  done  by  /'  will  be  Px'Zna.     During  the  same 


Fig.67 


THE  aCHEW. 


205 


ig  iis  ii  >  or 
Iso  Magnus's 


a  right  cir- 
lit'li  tliere  is 
.  iiu'IiiK'd  at 
e  axis  of  the 


f 


B7 

counterpart 
<  cut  is  often 
lit  the  end  of 
d  to  the  nut. 
?ssure  in  the 
pvolve  only  ; 
fht  angles  to 
iroduced. 

)r  and  the 

P.  to  act  in 
r  and  at  the 
•e\v  to  have 
lave  moved 
radius,  and 
Ig  the  saniu 


time  the  screw  will  have  moved  in  the  direction  of  its  axis 
through  the  distance,  AB  —  •*::)•  tan  «,  /•  being  the  radius 
of  the  cylinder,  and  «  tiie  angle  which  the  thread  of  the 
screw  makes  with  its  base.  Then  as  this  is  the  direction  in 
which  the  resistance  is  encountered,  the  work  done  against 
the  resistance,  W,  is  W'i^r  tan  «.  Hence  if  no  work  is  lost 
the  eciuatiou  of  work  will  be 


F  X  "-iTTrt  =   ir  X  'Z-^r  tan  k. 


(1) 


That  is  the  power  i.s  to  the  weight  as  the  pitch  of  ttie  screw 
is  to  the  circumference  descril>e't  by  the  power. 

If  there  is  fricticm  between  tiie  thread  and  tiie  groove,  let 
R  he  the  normal  pressure  at  any  poitit,  p,  of  the  thread, 
and  /t/i'  the  friction  at  this  point,  then  tiie  work  done 
against  the  friction  in  one  revolution  is  }i)LR-i,nr  sec  «,  1/' 
denoting  the  sum  of  tiie  normal  reactions  at  all  points  of 
the  thread.     Hence  the  eipiation  of  work  is 


P2-rta  =  ]\'2Trr  tan  «  +  ii2nr  sec  a^E. 


(2) 


But,  for  the  e(iuilibrium  of  the  screw,  resolving  parallel 
to  the  axis,  we  have 

W  =  i:  (^  cos  «  —  /«/?  sin  «), 

W 


therefore 


i:i?  = 


cos  rt  —  ^  sin  « 


which  in  (3)  gives 


or 


ur  sec  «  W 

j'li  =  ]\ r  tan  «  +  -—  -  -.r.j;r~'> 

cos  re  —  /*  sm  a 
Pa  ■-  Wr  tan  («  +  <}>}, 


(3) 


(p  Iteing  the  angle  of  friction, 


ita 


ao6 


PKoyy's  i>[FFEi{i:.\'rrM,  schew. 


129a.  Prony's  DiflFerential  Screw.— If  h  denoto  flie 
pitch  of  a  ticrew  (1)  ljL'C(>mL'.s 

^Pna  =  Wh, 

wliicli  expresses  the  reliitioti  between  /'  and  W,  when  fric- 
tion is  neglected.  Therefore  the  meehanieai  advantage  is 
gained  by  nniking  tlie  pitch  very  small.  In  some  cas-s, 
however,  it  is  desirable  that  the  screw  should  work  at  fair 
speed,  us  in  ordinary  bolts  and  nuts,  and  then  the  pitch 
must  not  bo  too  small.  In  cases  where  the  screw  is  used 
specially  to  obtain  pressure,  as  in  screw-presses  for  cotton, 
etc.,  we  do  not  care  for  speed,  but  only  for  jtressure.  But 
in  practice  it  is  impossible  to  get  the  jjitch  very  small  from 
the  fact  that  if  the  angle  of  inclination  is  very  Hat,  the 
threads  run  so  near  each  other  as  to  be  too  weak,  in  which 
ca<e  the  screw  is  apt  to  "strip  its  thread,"  that  is,  to  tear 
bodily  out  of  the  hole,  leaving  the  thread  behind. 

Where  very  great  pressure  is  rcpiired  a  differential  nut- 
/lole  is  resorted  to.  Let  the  screw  work  in  two  blocks, 
A  and  15,  the  first  of 
which  is  fixed  and  the 
second  movable  along  a 
fixed  groove,  n  ;  and  let 
h  be  the  ])itch  of  the 
thread  which  works  in 


K  wM 


Fig.  68 


the  block,  A.  and  //'  the  pitch  of  the  thread  which  works 
in  the  block  H.  Then  one  revolution  of  the  screw  impresses 
two  opposite  motions  on  the  block,  li,  one  equal  to  //  in  the 
direction  in  which  the  screw  advances,  and  the  other  eipial 
to  //'  in  the  opposite  direction.  If  then  the  block.  H,  is 
connected  with  the  resistance  W,  we  liave  by  the  principle 
of  work 

2Pna  =  II  (/*  —h')- 
itlld  the  rc((uisite  power  wjll  bo  diminished  by  diminishing 


I 


mmM 


EXAMPLES. 


207 


(Icnoto  file 


,  wlion  fric- 

itl vantage  is 
some  cas-'s, 
voi'k  at  fair 
II  tlie  pitch 
crew  is  used 

for  cotton, 
ssiiro.     But 

small  from 
M-v  flat,  the 
ik,  ill  which 
it  is,  to  tear 
1. 

reiitial  nut- 
two   blocks. 


hicli  works 
w  impresses 
to  It  ill  the 
oilier  e(|iial 
ilock.  H,  is 
e  |iriiiciple 


iminishing 


I 


//  _  //'.  By  means  of  this  screw  a  comparatively  small 
pressure  may  be  made  to  yield  a  pressure  enormously 
<rieater  in  magnitude. 

EXAMPLES. 

1,  A  lever  10  ins.  long,  the  weight  of  which  is  4  lbs.,  and 
acts  at  its  miildle  point,  balances  about  a  certain  point 
when  a  weight  of  G  lbs,  is  hung  from  one  end;  find  the 
lioiiit.  Ans.  2  ins.  from  the  end  where  the  weight  is. 

•I.  A  lever  weighing  S  Uis.  balances  at  a  point  3  ins.  from 
one  end  and  9  ins.  from  tlie  other.  Will  it  continue  to  bal- 
ance about  that  point  if  e(pial  weights  be  suspended  from 
the  extremities  ? 

3.  A  beam  whose  length  is  13  ft.  balances  at  a  point  2  ft. 
from  one  end  :  but  if  a  weight  of  100  lbs.  be  hung  tVom  the 
other  end  it  balances  at  a  point  3  ft.  from  that  end ;  find  the 
weight  of  the  beam.  Ans.  25  lbs. 

-1.  A  lever  T  feet  long  is  supported  in  a  horizontal  jiosi- 
tion  by  props  placed  at  its  extremities  :  find  where  a  weight 
of  28  lbs.  must  be  placed  so  (hat  the  pressure  on  one  of  the 
projis  may  be  8  lbs.  Ans.  Two  IVet  from  the  end. 

5.  Two  weights  of  i2  lbs.  and  8  lbs.  respectively  at  the 
ends  of  a  horizontal  lever  10  feet  long  lialance :  find  how 
far  the  fulcrum  ought  to  be  moved  for  the  weights  to  bal- 
ance when  each  is  increased  by  2  lbs.      Atis.  Two  inches. 

(i.  A  lever  is  in  equilibrium  under  the  action  of  the  forces 
/'and  (),  and  is  also  in  ei|uilibrium  when  P  is  trebled  and 
<,>  IS  increased  by  0  lbs.:  find  the  magnitude  of  Q. 

Ans.  3  lbs. 

;.  In  a  lever  of  the  first  kind,  let  the  power  bo  217  Uis  . 
the  weight  ivT)  lbs.,  and  the  angle  between  them  12(J  . 
Fiiui  the  [iressure  on  the  fulcrum,  Ans.  622.7  lbs. 


mm 


i 


308  EXAMPLES. 

8.  If  the  power  ami  weiglit   in  a  strai<?lit  lover  of  tho  f 
first  kind  be  17  lbs.  and  ^i  lbs.,  and  make  witli  "ach  other 
an  angle  of  70°;  find  the  pressure  on  tlie  fulcruni. 

A  lis.  39  lbs. 

9.  The  length  of  the  beam  of  a  false  balance  is  .'5  ft. 
9  ins.  A  body  placed  in  one  scale  balances  a  weight  of 
9  lbs.  in  the  other  ;  but  wiien  placed  in  tbc  other  scale  it 
balances  4  lbs.;  recn'ired  tiio  trne  weight,  W,  of  the  bod} 
and  the  lengths,  a  and  b,  of  the  arms. 

xins.    W  =  G  lbs.;  a  =  1  ft.  U  ins.;  b  =  i  ft.  3  ins. 

10.  If  a  balance  be  false,  l)aving  its  arms  in  the  ratio  of 
15  to  16,  find  iiow  much  ]m-  lb.  a  customer  really  ])avs 
for  tea  Avhich  is  isold  to  him  from  the  longer  arm  at  3s.  9'd. 
per  lb.  A)is.  4s.  peril). 

11.  A  straight  uniform  lever  whose  weight  is  50  lbs.  and 
length  0  feet,  rests  in  e(|uilibriuin  on  a  fulcrum  when  a 
weight  of  10  lbs.  is  suspen(le<l  from  one  extremity:  find  the 
position  of  the  fulcrum  and  the  i)ressure  on  it. 

Ans.  2^  ft.  from  the  end  at  which  10  lbs.  is  suspended ; 
60  lbs. 

12.  On  one  arm  of  a  false  balance  a  body  weighs  11  lbs.; 
on  the  other  17  lbs.  3  oz.;  what  is  the  true  weight? 

Ans.  13  lbs.  12  oz. 

13.  A  bent  lever  is  composed  of  two  straight  nnifV)rm 
rods  of  the  same  length,  inclined  to  each  other  at  120°,  and 
the  fulcrum  is  at  the  i)()iMt  of  intersection  :  if  the  weight  of 
one  rod  be  double  that  of  tlu'  other,  show  that  the  lever  will 
remain  at  rest  with  the  lighter  arm  horizontal. 

14.  A  uniform  lever.  /  feet  long,  has  a  weight  of  W  lbs., 
suspended  from  its  extremity:  find  the  position  of  the  ful- 
crum when  the  long  end  of  tlie  lever  balances  the  short 


li'vor  of  tho 

li  "acli  other 

nil. 

IS.  'M  lbs. 

nee  is  3  ft. 
a  weight  of 
her  scale  it 
of  the  bo(J} 

i  ft.  3  ins. 

the  ratio  of 
really  jiays 

•ni  at  3s.  9d. 

4s.  per  11). 

50  lbs.  and 
iim  when  a 
ty:  find  the 

suspended ; 


ghs  11  lbs.; 

ht? 

bs.  12  oz. 

ht  uniform 
it  1-^0°,  and 
le  weight  of 
lie  lever  will 


of  H'lbs., 

of  the  ful- 

!8  the  short 


EXAMPLES. 


209 


end  with  the  weight  attached  to  it,  supposing  each  unit  of 
length  of  the  lever  to  be  w  lbs. 

^"*-  ;rTTi^-7-  7-^.  '8  the  short  arm. 
i{\\  +  Iw) 

15.  A  lever,  I  ft.  long,  is  balanced  when  it  is  placed  upon 
a  prop  ^  ofits  length  from  the  thick  end;  when  a  weight 
df  \V  11)S.  is  suspended  from  the  small  end  the  prop  must 
be  shifted  j  ft.  towards  it  in  order  to  maintain  equilibrium ; 
required  the  weight  of  the  lever.  Ans.   \W. 

16.  A  lever.  I  ft.  long;  is  balanced  on  a  prop  by  a  weight 
of  \V  lbs.;  first,  when  the  weight  is  suspended  from  the 
thick  end  the  prop  is  a  ft.  from  it;  secondly,  when  the 
weight  is  suspended  from  the  small  end  the  prop  is  b  ft. 
from  it ;  required  the  weight  of  the  lever. 

Ans.   1 V--— rr  Ihs. 

I  —  {a  +b) 

17.  The  forces,  P  and  W,  act   at   tho   arms,   a  and  b, 

respectively,  of  a  .>^tr;iight  lever.     When  P  and   W  make 

angles  of  30°  and  90"  with  the  lever,  show  that  when  etpii- 

2b  W 
librium  takes  place  P  =     

18.  Supposing  the  beam  of  a  false  l)a1anco  to  be  uniform. 
a  and  b  the  lengths  of  the  arms,  /'  and  Q  the  apparent 
weights,  and  If  the  true  weight ;  when  the  weight  of  the 
beam  is  taken  into  account  show  that 


a 
b 


P-  W 


11" -<2 

19.  If  «  be  the  length  of  the  .'short  arm  in  Ex.  14,  what 
must  be  the  length  of  the  whole  lover  when  equilibrium 

lakes  place?  /aflir""T 

A  U.S.  a  +  \/    -    -  +  «". 

'M.   A  man  whose  weigiit  is  140  llis.  is  jnst  able  to  sup- 
port a  weight  that  hangs  over  an  axle  of  (i  ins.  nidius,  by 


5ilO 


EXAMI'IjES. 


liaiigiiig  to  the  rope  tluit  passes  over  tlie  corresponcllng 
wlu'c'l,  the  (iiameter  of  wliieli  is  4  ft,;  tiud  tlie  weiglit  &\\\)- 
I'orted.  Ans.  500  lbs. 

^'1.  If  tlie  diirerenoe  between  the  diaiiicter  of  a  wheel  -Mxi 
the  diameter  of  the  axk'  be  six  times  tlie  radius  of  the  axK\ 
tiiid  tlie  greatest  weigiit  that  euu  be  sustained  by  a  loree  of 
<•*>  ll>s-  An.'i.  240  lbs. 

'Z'i.  If  the  radius  of  tiic  wheel  is  three  times  that  of  the 
axle,  and  the  string  round  the  wiieel  ean  sup])ort  a  weight 
of  40  lbs.  only,  find  tiie  greatest  weight  that  can  be  lifted. 

Alls,    im  lbs. 

2:?.  What  force  will  be  required  to  work  the  handle  of  a 
windlass,  the  resistance  to  be  overcome  being  115G  lbs.,  the 
radius  of  the  axle  being  six  ins.,  and  of  the  handle  2  ft. 
yi"*-?  Ann.  2 10. Id  lbs. 

24..  Sixteen  sailorB,  exerting  each  a  force  of  29  li)s.,  push 
u  capstan  with  a  length  of  lever  ecpial  to  8  ft.,  the  radius  of 
the  capstan  being  1  ft.  2  ins.  Find  the  resistance  which 
this  fon  e  is  capable  of  sustaining. 

Ans.   1  ton  8  cwt.  1  (jr.  ITf  lbs. 

25.  Supposing  them  to  have  wound  the  roi)e  round  the 
capstan,  so  that  it  doul)les  back  on  itself,  the  radius  of  the 
axle  is  thus  increas"!  by  the  thickness  of  the  rope.  If  this 
be  2  ins.  how  much  wdl  the  power  of  the  instrui7ient  be 
diminished.  Ans.   By  J^,  or  12^  jter  cetit. 

20.  The  radius  of  the  axle  of  a  capstan  is  2  feet,  and  six 
men  push  each  with  a  force  of  one  cwt.  on  spokes  5  feet 
long;  find  the  tension  they  will  bo  able  to  produce  iji  the 
rope  which  leaves  the  axle.  Ans.   15  cwt. 

27.  The  difference  of  the  diameters  of  a  wheel  and  axle 
is  2  feet  0  inches  ;  and  the  weight  is  eijual  to  six  limes  the 
power  ;  find  the  radii  of  the  wheel  and  the  iix'e. 

Ans.  18  ins. ;  3  ins. 


i 


*j 


Tcspomllng 
veight  sup- 
5U0  lbs. 

I  wheel  iini 
)f  the  iixle. 
■  a  t'oree  oi 

that  of  the 
t  a  W('i<,'lit 

l)e  lifted. 

l-iO  lbs. 

laiuUe  of  a 
50  lbs.,  the 
laiidlc  2  ft. 
C.T5  lbs. 

)  I1)S.,  push 

0  radius  of 
iuco  which 

irt^lbs. 

round  the 
litis  of  the 
le.  If  this 
rument  be 
»er  eetit. 

?f,  and  six 
ikes  i)  feet 
lice  in  the 
15  ewt. 

1  and  axle 
limes  the 

. ;  3  ins. 


i 


«/ 


EXAMl'l.KS. 


211 


28.  If  the  radius  of  a  wheel  is  4  ft.,  and  of  the  axle 
K  ins.,  lind  the  power  that  will  balance  a  weight  of 
SOU  lbs.,  the  thickness  of  the  rope  coiled  round  the  axle 
being  one  inch,  the  power  acting  without  a  rope. 

Ans.  88.5-1  lbs. 

29.  Two  given  weights,  P  and  Q,  hang  vertically  from 
two  points  in  the  rim  of  a  wheel  turning  on  an  axis; 
find  the  pi)sition  of  the  weights  when  equilibrium  takes 
place,  sni)posing  the  angle  between  the  radii  drawn  to 
the  points  of  suspension  to  be  •JO'',  and  that  0  is  the 
an<de    wliich    the    radius,    drawn    to    i"s   point   of    sus- 


pension, makes  witli  the  vertical. 


Ans.  tan  0  = 


9. 
r' 


30.  What  weight  can  l)e  fUi»ported  on  a  plane  by  a  hori- 
zontal force  of  10  lijs.,  if  the  ratio  of  the  height  to  the  base 
is  3  v  Ans.   13-i  lbs. 

31.  The  inclination  of  a  plane  is  30°,  and  a  weight  of 
10  lbs.  is  supported  on  it  by  a  string,  bearing  a  weight,  at 
its  extremity,  which  passes  over  a  smooth  pulley  at  its 
summit ;  find  the  tension  in  the  string.  Ans.  5  V)S. 

32.  The  angle  of  a  plane  is  45°  ;  what  weight  can  be 
supported  on  it  by  a  horizontal  force  of  3  lbs.,  and  a  force 
of  4  lbs.  parallel  to  the  plane,  both  acting  together. 

Ans.  3  +  4  V^  lbs. 

33.  A  body  is  supported  on  a  plane  by  a  force  parallel 
to  it  and  e<|ual  to  ^  of  the  weight  of  the  body ;  find  the 
ratio  of  the  height  to  tlie  base  of  the  plane. 

Ans.  1  :  2\/0. 

34.  One  of  tlie  longest  inclined  planes  in  the  world  is 
the  road  from  Lima  to  (Jallao,  in  S.  America  ;  it  is  0  miles 
long,  and  the  fall  is  511  ft.     Calculate  the  inclination. 

A71S.  55'  2T",  or  1  yard  in  03. 


212 


EXAMPLES. 


35.  If  the  forc'u  rcquiri'd  to  draw  m  wiifroii  on  ji  horizontal 
roiul  he  ^S^th  part  oi"  the  weight  of  the  wagon,  wl'at  will  l)e 
the  force  required  to  draw  it  up  a  hi'l,  the  sloi»c  of  which 


is  1  in  \o. 


Alts.  Tj.Vi^''  l"^'"*^  of  tlie  weight. 


3G.  If  the  force  recpiired  to  draw  a  train  of  cars  on  a 
level  railroad  be  jj-ftth  i)art  of  tiie  loud,  lind  the  force 
required  to  draw  it  up  a  grade  of  1  in  5(1. 

A  U.S.  jg^Jys*'^  P'lJ't  of  the  load. 

37.  What  force  is  required  (negleciing  friction)  to  roll  a 
cask  weighing  !»(i4  lbs.  into  a  cart  3  ft.  high,  by  means  of  a 
j)lank  14  ft.  long  resting  against  (he  cart. 

^liis.  'i'he  force  must  exceed  2(»(i^  lbs. 

38.  A  body  is  at  rest  on  a  sniootb  inclined  plane  when 
tlic  power,  weight  and  normal  ])ressure  are  18,  ^0,  and 
12  lbs.  respectively;  find  the  inclination,  «.  of  th.  i)lane  to 
the  horizon,  and  the  angle,  0,  which  the  dircciiou  of  the 
power  makes  with  the  plane. 

An.s.  u  z^  37"  21'  20" ;  0  =  28°  4G'  54". 

30.  If  the  power  which  will  sujiport  a  weight  when  ict- 
ing  along  the  piane  be  half  that  which  will  do  so  acting 
horizontally,  find  the  inclination  of  the  plane.  Aiis,  (',0°. 

40.  A  jwwei-  /'acting  along  a  plane  can  support  IT,  and 
acting  hoii>;ontally  can  support  x  ;  show  (hat 

41.  A  weight  ll' would  he  supported  by  a  jiower  P  act- 
ing horizontally,  or  by  a  power  Q  acting  i)arallel  to  the 
l)lane ;  show  that 

Qi  />..  ^    1(1! 

42.  The  base  of  an  indiiu'd  plane  is  8  ft.,  the  height 
(1  ft.,  and  \y  =  10  Ions;  reipiircd  /*  and  (he  normal 
pressure,  N,  on  the  plane. 

J  A-.,  r  —  G  tous;  N  =  S  tons. 


i 


•J 


EXAMt'LHS. 


213 


I  horizontal 
I'lit  will  be 
B  of  which 
?  weight. 

'  curs  on  a 
I  tlio  force 

tlio  load. 

i)  to  roll  a 
means  of  a 

■imii  lbs. 

)iane  when 
8,  2G,  ami 
K'  plane  to 
ion  of  the 

46'  54". 

when  ict- 
[)  so  acting 
//.<;.  00°. 

>rt  ir,  and 


•er   P  acl- 
llel  to  the 


he  Iieiglit 
ic    iiorniiil 

8  tons. 


i 


I 


43.  A  weight  is  supported  on  an  inclined  plane  Ijy  a 
force  whose  direction  is  inclined  to  liie  plane  at  an  angle 
of  30"  ;  when  the  inclination  of  the  plane  to  the  horizon  is 
30',  show  that  11'=  P  ^/d. 

44.  A  man  weighing  IhQ  lbs.  raises  a  weight  of  4  cwt.  by 
a  system  of  fonr  movable  pulleys  arranged  according  to  the 
second  system  ;  what  is  his  pressure  on  the  ground  ? 

A)is.  \n  lbs. 

45.  What  power  will  be  required  in  the  sk  und  system 
with  four  movable  jJuUeys  to  sustain  u  weight  of  IT  tona 
VZ  cwt.  Atis.    I  ton  -i  cwt. 

40.  Two  weights  hang  over  a  pulley  fixed  to  the  summit 
of  a  smooth  inclined  plane,  on  which  one  weight  is  sup- 
ported, ami  for  every  3  ins.  that  one  descends  the  other 
rises  2  ins. ;  find  the  ratio  of  the  weights,  ami  the  length 
of  the  plane,  the  height  being  IS  ins.    Ans.  2  :  3  ;  27  ins. 

47.  If  W  —  330  lbs.  and  P  —  42  lbs.  in  a  combination 
of  pulleys  arranged  according  to  the  first  system,  how  many 
movable  i)ulleys  are  there  ?  An^.  4. 

48.  In  a  system  of  i)iilleys  of  the  th-rd  kind  in  which 
there  arc  4  cords  attached  to  the  weig*  ',  determine  the 
weight,  \y,  suppofV'd,  and  the  strain  on  ttie  fixed  ]nilley. 
the  power  being  100  lbs.,  and  the  weight,  w,  of  each 
pulley  5  lbs. 

Ans.  \\  —  157'  +  \\w  =  1555  lbs. ;  Strain  =  Ul'  +  lbto 
=  1075  lbs. 

4!).  In  a  system  of  pulleys  of  the  third  kind,  there  arc 
2  movalde  pulleys,  each  weighing  2j^  lbs.  What  power  is 
required  to  ii.pport  a  weight  of  0  cwl.  ?     .Iw.v.   !)4.57  lbs. 

50.  Kind  the  power  thai  will  support  a  weight  of  KM)  His. 
by  means  ((f  a  system  of  -I  pulleys,  the  strings  being  all 
attached  to  the  weight,  and  each  pulley  weighing  1  II). 

Alls.  5|J  lb.s. 


^m 


2U 


KXAMVLES. 


51.  Tlie  lircumferoiK'e  of  tho  circle  corresponding  io  tho 
point  of  upplicution  of  P  is  6  feet;  find  how  many  turns 
the  screw  must  make  on  a  cylinder  5J  feet  long,  in  order 
th.it  ll'may  oe  ecpnil  to  144/'.  Aiif<,  48. 

b'Z.  Tiie  distance  l)ctw('cn  two  consecutive  threads  of  a 
screw  is  a  quarter  of  an  inch,  and  tiie  longtli  of  tiio  powci  * 
arm   is  a  feet  ;  lind  what  weight  will  be  susiained   l»y  a 
power  of  1  lb.  Ans.  480tt  lbs. 

53.  IIow  many  turns  must  be  given  to  a  screw  formed 
upon  a  cylinder  whose  length  is  10  ins.,  and  circumference 
5  ins.,  that  a  power  of  'i  ozs.  may  overcome  a  pressure  of 
lOOozs.?  Ans.   100. 

54.  A  screw  is  nuide  to  revolve  by  a  force  of  2  lbs. 
applied  at  the  entl  of  a  lever  3.5  ft.  long;  if  the  distance 
between  the  threads  be  \  in.,  what  i)rcssure  can  be  pro- 
duced? Ans.  1)  cwts.  1  ([r.  20  lbs. 

55.  The  length  of  the  power-arm  is  15  indues;  find  the 
distance  between  two  consecutive  threads  of  the  screw, 
that  tlio  mechanical  advantage  may  be  30.       Ans.  n  ins. 

50.  A  weight  of  ir  jiounds  is  suspended  from  the  idock 
of  a  single  nu)val)le  pulley,  and  the  end  of  the  cord  in 
which  the  powt'r  acts,  is  fastened  at  the  dislance  of  />  ft. 
from  (lie  fulcrum  of  a  ]iori7,ontal  lever,  a  ft.  long,  of  the 
seconil  kind  ;  lind  the  force.  /'.  which  must  be  applied  per- 
pendicularly ut  the  extremity  of  the  lever  to  sustain  IT. 

Am.  /'  =    .    • 


\ 


57.  Tn  a  steelyard,  the  weight  of  the  beam  is  10  lbs.,  and 
the  distance  of  its  centre  of  gravity  from  (he  fulcrum  is 
'.'  ins.,  lind  where  a  weight  of  4  lbs,  must  be  placed  Io  bal- 
ance it  A  ns.  At  5  ins. 


KXAMl'LKS. 


215 


ding  lo  tho 
11  iiiy  turns 
g,  in  order 
Ann.  48. 

iroads  of  :i 
tiio  powi'i  * 
lined  l»y  a 
■180TT  lbs. 

rew  formed 
3umferenco 
pressure  of 
Iws.   100. 


5H.  A  body  whose  weigiit  is  \/'Z  lbs.,  is  ])luoed  on  a  rough 
phme  inclined  to  the  horizon  at  an  angle  of  45°.     The  co- 

ctticientof  friction  being       ,  find  in  wliat  direction  a  force 

of  (v'lJ  —  I)  ll'S.  must  act  on  the  body  in  order  just   Id 
sui)port  it.  .1//.S'.  At  an  angle  of  30°  to  the  jjlane. 

f)!).   A  rough  plane  is  inclined  to  the  horizon  at  :i:;  angle 
of  00°  ;  find  tlie  magnitude  and  t lie  direction  of  the  least 
for-e  which  will  prevent  a  l)ody  weighing  1(10  lbs.  from  slid- 
ing down  the  i)lane,  the  coellicieut  of  friction  being  — — • 
Ans.  50  lbs.  inclined  at  30°  to  the  plane. 


e  of  2  lbs. 
lie  distance 
■an  be  pro- 
r.  20  lbs. 

s;  find  tho 
the  screw, 
(s.  TT  ins. 

I  the  i)lock 
he  cord  in 
ice  of  h  ft. 
ong,  of  the 
ipplied  per- 
tain U'. 
,,  _  Wb 

~  aa  * 


10  lbs.,  and 
fulcrum  is 
ccd   lo  l;al- 
At  5  inu. 


CHi^  PTER  VIII. 


THE  FUNICULAR*  POLYGON— THL:  CATENARY 
ATTRACTION. 

130.  Equilibrium  of  tho  Funicular  Polygon.— If  u 

cord  wliose  weight  is  lu'glected,  is  .suspetidud  IVoin  two  fixed 
points,  .1  and  B,  and  if  a  series  of  weiglits,  1\,  P^,  J\, 
etc.,  be  suspended  from  the  given  points  Q^.  Q.^,  (>.,.  e'c, 
tlie  cord  will,  when  in  eqiiilihrinm,  form  a  polygon  in  a 
vertical  plane,  which  is  called  the  Funicular  Polygon. 

Let  the  tensions  along 
the  successive  portions 
of  the  cord,  .'!(;>,,  Q^Q^, 
Q^Qi,  etc.,  be  respec- 
tively ',,  T^,  7\,  etc., 
anu  let  0^,  0^,  0^,  etc., 
l)e  the  inclinations  of 
these  portions  to  the 
horizon.  Then  Qf  is 
in  equilibrium  under  the  action  of  three  forces  viz.,  /',, 
acting  vertically,  7',.  the  tension  of  the  cord  .!(),,  ami  T^, 
the  tension  of  (J^  Q^.     Resolving  these  forces  we  have, 

for  horizontal  forces,         7',  cos  0,  —  7'j  cos  0^=0,  (1) 

for  vertical  forces,  /',  +  T^  sin  6„  —  7\  sin  0,  =  0,  (2) 

In  the  same  way  for  the  point  Q^  we  have, 

for  horizontal  forces,         7',,  cos  6.,  —  7'.,  cos  0,,  :=  0,  (3) 

for  vertical  forces,  I'...  +  7'.,  sin  0^        'l\  sin  fl„  —  (i,  (4) 


Fig.60 


♦  The  term,  F^inlciilar.  hno  rc'lcn^ict^  uloiic  to  llio  cord,  iiinl  Iiiih  no  iinThiiniciil 
fi((iilflcaiice. 


TENARY 

lygon. — If  a 
)in  two  fixed 
P     /'     /' 

■'    1'   •'   2'   ^    3' 

olytjoii  in  u 
olygon. 


T«i 


p, 


R 


^ 


/p. 


J^ 


!0s  viz.,  /',, 
Qi,  and  7'g, 
■0  have, 

!    =   0,        (1) 

I  =  0,     (2) 


.  =  0,      (3) 
J  -  <'-      (4) 


H  no  nu'obiiiiical 


EQUILUililUM  OF  THE  FUMVLLAH   rOLYGOX.      217 

Hence  from  (1)  and  (3)  we  have 

2\  cos  0,  =  7^2  cos  02  =  7^3  cos  03  =  etc. , 

that  is,  the  horizontal  components  of  the  tensions  in  tlie  dif- 
ferent portions  of  tlte  cord  are  constant.  Let  this  constant 
be  denoted  by  T;  then  we  have 


T,  = 


T 


.     T    —  = 

'        *  ~  cos  0, 


*  COS   0 


cos  0, 

which  in  (3)  and  (4)  give 

I\  +  rtan  flj  —  ytan  0,  =  0, 
I\  +  T  tan  03  -  r  tan  0,  =  0, 

and  from  (5)  and  (0)  we  have 


;  etc., 


(6) 


tan  0,  =  tan  0g  +  -J, 

and 

p 

tan  0j  =  tan  03  +  -  '  • 

Similarly 

/> 

tan  03  =  tjin  0^  4-    ,y?, 

and 

p 

tan  0.  =  tan  0.  +  -  *, 

(7) 


etc.,  etc. 

If  we  suppose  tiie  weights  P,,  1\,  etc.,  each  equal  to  IK, 
(7)  becomes 


tan  0, 


tan  0g  =  tan  0j  —  tan  0.  =  tan  03  —  tan  0^ 


(«) 


Hence,  tlie  tangents  of  the  successive  inclinations  form  a 
series  in  Arithmetic  Progression.     In  the  figure  0,  =  0, 
10 


5J18    coysTurcTioy  of  the  FVSivrLAii  voiacos. 


tan  Q^ 
tan  0g 


^,;     tan  03  =  -^\ 


3  IF 


4ir 


(f') 


™- ;  tan  ^i  =  -^  ;  t'te 


i,. 


131.  To  Construct  the  Funicular  Polygon  when 
the  Horizontal  Projections  of  the  successive  Por- 
tions of  the  Cord  are  all  equal.  -I  At  (),,Q^.  QiQi^'ls'li-' 
'Ii'/\'  ^'t''v  ''I'  nil  of  con.stiuit  longtli  =  a,  uiul  lei  Q^q^  =  r. 
Then  since  by  (!l)  of  Art, 
]:}(),  the  tan<,H'iits  of  0^,  d.^, 
(K,  Oi,  ete.,  are  as  1,  '2,  li, 
4,  etc.,  we  have 


sl'^ 


q/0 

III 


<h  'I'  'J< 


Q,H   =  WQ^q^  =  ■36';  etc.  Fig.70 

Hence,  takinj:;  the  middle  point.  O.  of  the  horizontal 
l)ort'ioi).  Qf,Qn,  as  orif^in,  and  the  horizontal  and  vertical 
lines  tlirouiih  it  as  axes  of  x  and  ?/,  the  co-ordinates  of  (>, 
arc  (|rt,  c) ;  those  of  Q„  are  (|r/.  Wc) ;  those  of -^j  are  {la. 
Be),  and  those  of  the  ni\\  vertex  from  Q^  arc  evidently 

2n  +  1 


X  = 


2 


n  (//  4-  1) 


Eliminating  ti  from  these  eciuations  we  get 

-   4- 
c 


3? 


_  'iahj      a* 

''  "^  4 


(1) 


which,  being  independent  of  «,  is  satisfied  by  all  the  ver- 
tices inditfcrently,  and  is  therefore  the  e(|iiation  of  a  curve 
passing  through  all  the  veilices  of  tiie  polygon,  iind 
denotes  a  |)araliola  whose  axis  is  the  verlicid  line.  OV,  iiiid 

whose  vertex  is  vertically  below  (>  at  a  di-tauce  =     • 

'The  shorter  the  distances  ^,>,,  Vii-  QtQi^  ^^^'•-  ^'"'  >ii"i*' 
nearly  does  the  funicular  j>i)ly;/iiii  coincide  with  the  para 
bolic  curve. 


)V,O.V. 


(0) 


jon  when 
sive  Por- 

''i^'/,v'/.•^'/2' 


Q^i 

3  =  '■■ 

/ 

A 

i 

/ 

n 

H 

X 

horizontal 
11(1  vertical 
latt's  of  Q^ 
Qi  are  (Jrt, 
duntly 


(1) 

ill  the  vor- 
1  of  a  curve 
IviTdii,   and 

•".  OV.  and 

r 

~  s' 

.    tllC    IMiil'i' 

li  the  para 


com)  sci'i'ouTiyo  load. 


2l!t 


132.  Cord  Supporting  a  Load  Uniformly  Dis- 
tributed over  the  HorJroataL — If  tlie  iiuiiil)er  of  vertices 
of  the  i)oly^foii  be  very  great,  and  the  suspended  weiglits  ail 
('(|iial  so  tliat  the  load  is  (iistril)utcd  uniformly  along  the 
.straight  line,  FE,  the  parabola  whicii  j)asses  through  all  llio 
vertices,  virtiii.lly  coincides  with  tlu'  cord  or  chain  forming 
the  jiolygon,  and  gives  the  tigure  of  the  Situprnsioii  lindoi'. 
In  this  bridge  the  weights  suspended  from  the  successive 
portions  of  the  chain  are  the  weights  of  ecpial  ]K)rtio!is  of 
the  flooring.  The  weight  of  the  chain  itself  aiul  the 
weights  of  the  sustaining  bars  are  neglected  in  coiin)arisou 
with  the  weight  of  flooring  and  the  load  which  it  carries. 


Let  the  span,  AB.  =  2a,  and  the  lieight,  OD,  =  h. 
'i'hon  the  equation  of  the  parabola  referred  to  the  vertical 
and  horizontal  axes  of  .e  and  y,  respectively,  through  0,  is 


y2  —  inix, 


(1) 


■lin  being  the  parameter. 

Hecause  the  load  between  0  and  A  is  uniformly  dis- 
tributed over  the  horizontal,  OE.  its  resultant  bisects  OK 
ai  (";  therefore  the  tangents  at  A  and  O  intersect  at  C 
i.\rt.  (;■.'). 

fmiu  ( I)  Ave  have 

(///  _  "iiii  _  y 
dx  ~   f  ~  U' 


220  COKD  SlPl'ORTING    LOAD. 

which  is  the  tangent  of  tiic  inclination  of  tlic  curvo  at  any 

point  {x,  y)  to  tiie  axis  of  ./•.     Ilonco  the  tangent  at  the 

point  of  support,  A,  makes  with  the  horizon  an  angle,  «, 

2/t 
whose  tangent  is  "   ,  which  also   is  evident   from  the  tri- 

angle  ACE. 

Let  ir  be  the  weight  on  the  cord  ;  then  1 11'  is  the  weight 
on  OA,  and  therefore  is  the  vertical  tension,  V,  at  A.  Then 
the  tiiree  forces  at  A  are  tlie  vertical  tension  V  =  |U',  the 
total  tension  at  the  end  of  the  cord,  acting  along  the 
tangent  AC,  and  the  horizontal  tension,  T,  which  is  every- 
where the  same  (Art.  i;JO).  Hence,  by  the  triangle  of 
forces  (Art.  81)  these  forces  will  be  represented  l)y  the 
three  lines,  AE,  AC,  CE.  to  which  their  directions  are 
respectively  parallel ;  therefore  we  have  for  the  horizontal 
tension 

W 


T=  AE  cot  «  =  W 


and  the  total  tension  at  A  is 


W 


in 


EXAMPLE. 

The  entire  load  on  the  cord  in  (Fig.  71)  is  320000  lbs.; 
the  span  is  150  ft.  and  the  height  is  IT)  ft.;  lind  the  tension 
at  the  points  of  support  ami  at  the  lowest  point  and  also  the 
inclinati(m  of  the  curve  to  the  horizon  at  the  points  of 
support. 

tan  «  =  ''^'  =:  A;      .-.    «  =  21°  48'. 
(I 

The  vertical  tension  at  each  point  of  support  is 

r  —  i  weight  :      1(10000  lbs.; 


I 


^1^ 


curvo  at  any 

ugent  at  t!io 

an  angle,  «, 

roni  the  tri- 


is  the  weight 
at  A.  Thon 
'  =  i  W,  the 
g  along  the 
licli  is  every- 
.'  triangle  of 
ntcd  l)y  the 
irections  are 
le  horizontal 


320000  lbs.; 
I  tlie  tension 
:  and  also  the 
lie  points  of 


8'. 


IS 


f 


Tit/-:   C<tMM().\    CATi:XARi'. 

the  horizontal  tension  is 


m 


..  n 


T  —  W  ,,  ^  400000  lbs.: 


and  the  total  tension  at  one  end  is 

^  y-il^T^  =  430813  lbs. 

133.   The  Common  Catenary. — Its   Equation.— A 

catenary  is  the  curve  assumed  l)y  a  {)eri"uctly  llexible  I'ord 
when  its  ends  are  fastened  at  two  points,  A  and  B,  nearer 
together  than  the  length  of  the  cord.  When  the  cord  is  of 
constant  thickness  and  density,  i.  e.,  when  equal  portions  of 
it  are  equally  heavy,  the  curve  is  called  the  Cuvimon 
Catenary,  which  is  the  only  one  we  shall  consider. 

Let  A  and  B  be  the  fixed 
points  to  which  the  ends  of 
the  cord  are  attached  ;  the 
cord  will  rest  in  a  vertical 
plane  passing  through  A  and 
B,  which  may  be  taken  to  be 
the  plane  of  the  paper  Let 
C  bo  the  lowest  point  of  the 
catenary;  take  this  as  the 
origin  of  co-ordinates,  and 
let  the  horizontal  line 
through  C  be  taken  for  the 
axis  of  r,  and  the  vertical 
line  through  C  for  the  axis  of  y.  Let  {x,  y)  be  any  point, 
P,  ii  the  curve  ;  deiiote  the  length  of  the  arc,  C/'.  by  .s'  ; 
let  6'*  be  the  length  of  the  cord  whose  weight  is  etjual  to 
the  tension  at  0;  and  T  the  length  of  the  cord  whose 
weight  is  etiual  to  the  tension  at  /'. 


Y 

/ 

^ 

/ 

\ 

/* 

\ 

N 

i 

\ 

P 

\ 

^ 

X 

C 

Tl 

/ 

O 

/ 

X 

Fig. 72 


»  Tlie  weight  of  a  iiuit  of  lengtli  of  Uie  cord  beins  here  taken  no  tin'  unit  ci( 
wui);ht. 


^ 


iin 


a22  Tin:  com.ho.\  f.r/fcw.iA'i'. 

Tlioii  the  iirc,  ('/'.  aftor  it  lias  iissiimcd  its  pcm.aneiit 
form  of  0(|iiilil)riiiiii.  may  Ik"  considored  as  a  rigid  body 
kept  ,it  ri'st  hy  tlireo  forci'S.  \iz.:  (1)  7'.  (lie  tciisinn.  acliiig 
at  /'along  tlic  tuiigont.  (•^)  r,  the  horizontal  tension  at  the 
lowest  point  ('.  and  ('■))  the  'Veiglit  of  the  eord.  ('/'.  acting 
vertically  downward,  and  denoted  by  x.  Draw  /'7"  the 
tangent  at  /',  meeting  the  axis  of // at  7".  Then  by  the 
triangle  of  forces  (Art.  31),  these  forces  may  be  represented 
by  the  three  lines  FT',  \F.  7"X  to  which  their  directions 
are  respectively  parallel.     Therefore 

T'N  _  vveiglit  of  CP 
NF  '~   tension  at  G  ' 

dy       s  ,  . 

Differentiating,  substituting  the  value  of  ds,  and  reducing, 
we"  have 

\dx/        _  (te 
~  c 


^/^*m 


dv 

Integrating,  and  remembering  that  when  a;  =  0,  t-  =  0, 


we  obtain 


log 


where  e  is  the  Naperian  base.     Solving  this  ct|Uation  for 

/,  we  obtain 
ax 


!  porn.anent 
I  rigid  boijy 
isinii.  acliiii,' 
iisiiin  at  lli<> 
.  CI',  acliii.ir 

nv  rr  tlu- 

Tlu'ii  In'  tl)(' 

roprescnti'd 

lir  directions 


id  reducing, 


=  o,|  =  o. 


equation  for 


(I) 


Tin:  ((jMMit.X  r.iVA'.V.i//)-. 


2'i^ 


and   l»y  integration,  ol)serving  tliat  //  =  U    wiien   x  =  0, 
we  have 


y 


=  '.,  ((^  +'•')-  c, 


(5) 


wliieli  is  the  e(|ualion  recjnired.  We  may  simplify  tliis 
eijuation  by  nioving  the  origin  to  the  point,  O,  at  a  dis- 
tance e(|ual  to  c  below  C,  by  putting  //  —  c  for  y.,  so  that 
(3)  becomes, 


y  =  \\^  +'")' 


(3) 


which  in  the  eqtialion  of  the  ratonanj,  in  the  usual  form. 
'Hie  horizontal  line  through  0  is  called  the  directrix*  of 
Ihe  catenary,  and  O  is  called  the  oriyin. 

CoK.  1.— To  find  the  length  of  the  arc,  CP,  we  have 

=  V  1  +  iV''  -  ^~V  '/^'  fi"""^  (1)' 
=  i  (e^  +  e~')  dx ;  (4) 

^  -  .  <' j  (5) 


the  constant  being  =  0,  since  when  x  —  0.  s  7=  0. 
This  etiuation  may  also  be  found  ininicdiat-.ly  by  equa- 

ting  the  values  of  ;    in  (a)  and  (1;. 


dx 


*  8et^  Price's  Anal.  Meclis.,  Vol.  I,  p.  416. 


^ 


iU 


Tin:  coM.}fo.\  cATi-.'XAiir. 


Cor.  2.— Since  r  =  Of  is  tiic  length  of  tlie  cord  whose 
weight  is  ecjiial  to  tiie  tension  of  the  curve  at  tlie  lowest 
l)oint,  C,  it  follows  tliat,  if  the  half,  /W,  of  the  curve  were 
removed,  and  a  cord  of  length  r,  icud  of  the  same  tliickness 
and  density  as  llie  cord  of  the  eurve.  wrre  joined  to  the 
are  67*,  and  siisi)ended  over  a  smooth  peg  at  (',  the  curvf 
would  be  in  cquilihriuin. 

CoK.  3.— We  iiave  from  the  triangle,  PNT', 


tension  at  P 
tension  at  6' 


pjv 

FN' 


or 


T 

c 


^  =  ^  from  (3)  and  (4), 


that  is,  fhe  tension  at  nrii/  point  of  tlie  catenary  is  equal  to 
the  weifj/it  of  a  jtortion  of  llie  cord  whose  lenyth  is  equal  to 
tlie  ordinate  at  tliat  point. 

Therefore  if  a  cord  of  constant  thickness  and  density 
/langs  freely  over  any  two  smooth  i)egs,  the  vertical  por- 
tions which  hang  over  the  i)egs,  must  each  terminate  on 
the  directrix  of  the  catenary. 


Cor.  4. — From  (3)  and  (5)  we  have 

if  =  s^  +  c^, 

and  from  (0)  we  hove 

dy 


(fi) 


(7) 


At  the  point,  P.  draw  the  ordinate.  PM,  and  from  M 
the  foot  of  the  ordinate,  draw  the  perpendicular  3/7'.   Then 


PT 


yco,}frT^y'^, 


II'  cord  whoso 
iit  the  lowest 
e  curve  were 
me  thickness 
joined  to  tlie 
(',  the  curve 


ry  ?,«  equnl  to 
'A  is  equal  to 

and  density 
vertical  por- 
terminate  on 


(fi) 


(7) 

ind  from  M 
ir  AfT.   Then 


rUE   CUMMoy  CATENARY. 

which  in  (7)  gives 

PT  =  s  =  the  arc,  CP, 


225 


(B) 


and  since  f  =  PT' +  TJP,  we  have  from  (0)  and  (8) 

TM  =  c.  (^) 

Therefore  the  point,  T,  is  on  the  involute  of  the  catenary 
which  ori-inateK  from  the  curve  at  C,  TM  is  a  tangent  to 
this  involute,  and  77',  tiie  tangei.t  to  the  catenary,  is 
Tiormal  to  the  involute,  (See  C!alculus,  Art.  124).  As  TM 
i^  the  tangent  to  this  last  curve,  and  is  equal  to  the  con- 
stant quantity,  c,  the  involute  is  the  equitangeutial  curve, 
or  tractrix  (See  Calculi's  p.  357). 

By  means  of  (8)  an.  I  (9)  we  may  construct  the  on^rtwand 
(lireclrix  of  the  catenary  as  follows  :  On  the  iauf/ent  at  any 
point,  P,  measnre  of  a  hmitli,  PT,  equnlio  the  arc,  CP ; 
at  T  erect  a  perpendicular,  TM,  to  tlie  tanyent  meeting  ttie 
ordinate  of  P  at  M;  then  tlie  horizontal  line  tlirouyh  M  ts 
tlie  directrix,  and  itn  intersection  with  the  axis  of  the  curve 
is  the  origin. 

(Job.  5.— Combining  (2)  and  (5)  we  obtain 


therefore 


{y  +  f-Y  =  s^  +  <^> 
^  z=  y^  +  2cy. 


(10) 


The  catenary  possesses  many  interesting  geometric  and 
mechanical  properties,  but  a  discussion  of  them  would 
carry  us  beyond  the  limits  of  this  treatise.  The  student 
who  wishes' to  pursue  the  subject  further,  is  referred  to 
Price's  Anal.  Mechs.,  Vol.  I,  and  Mincliin's  Statics. 


® 


220 


SPHKRWAL   SHELL. 


133a.  Attraction  of  a  Spherical  Shell. — By  iho  law 

of  universal  gravitation  even  particle  of  matter  attracts 
every  other  i)artielc  with  a  force  that  varies  diirr/Iif  as  the 
mass  of  the  attracting  jiai'licle.  and  i?ir('r.srly  as  the  S([uare 
of  the  distance  between  the  particles. 

To  find  the  resultant  (ittrartioii  of  a  spherical  shell  of 
uniform  density  and  small  uniform  thickness,  on  a  par- 
ticle. 

(1)  Suppose  the  particle,  P, 
on  which  the  value  of  the 
attraction  is  required,  to  be 
outside  the  shell. 

Let  p  and  k  he  the  density 
ami  thickness  of  the  shell,  0 
its  centre,  and  M  any  ])article  of  it.  Let  OM  =  a, 
PM  —  r,  OP  =  c,  the  angle  MOP  =  d,<p  the  angle  which 
the  plane  MOP  m.  kes  with  a  fixed  jdane  through  OP. 

Then  vhe  mass  of  the  element  at  M  (Art.  88)  is 
ph  a^  sm  d  dd  d(f>.  The  attraction  of  the  whole  shell  acts 
along  0P\  the  attraction  of  the  elementary  mass  at  M  on 
P  in  the  diroctioji  PM 

pi-  a^  sin  0  dO  dtp  ^ 


thercforo  the  attraction  of  Jlf  on  P,  resolved  along  OP, 


pi-  n^  sin  0  dO  d(p  c  —  a  cos  8 


(1) 


We  shall  eliminate  0  from  this  equation  by  means  of 
r»  =  rt2  +  (,-2  _  2ac  cos  0 ; 


rdr  =  ac  sm  6  dO  \ 


^^ 


-Ry  die  law 
ter  attracts 
I'lljl  as  tli(> 
tlio  S(|uaro 


cdl  shell  of 
on  a  par- 


^»>^ 


OM  =  a, 

i>Sl«  which 
h  OP. 

^i-t.  88)   is 

slioll  acts 

«  at  M  on 


OP, 


(1) 


of 


Sl'lIKltWAL   SHELL. 


Sin  W«0  =:  , 

ac 


and 


c  —  a  cos  0 


substituting  these  values  in  (1),  the  attraction  of  il/ on /* 
along  PO 

oka  /,       c?  —  d\  ,    ,j 

To  obtain  the  rtsnltant  attraction  of  the  whole  shell,  wo 
take  the  0-.ntegral  l)etvveon  the  limits  0  and  2t,  and  the 
r-integral  between  c  —  a  and  c  +  u. 

Honce  the  resultant  attraction  of  the   hell  on  /'along  PO 


k'n^ka^  mass  of  the  shell 

___  _  ^^^ 


(3) 


Since  c  is  the  (^stance  of  the  point  P  from  the  centre  this 
shows  that  the  attraction  nf  thi'  shell  on  the  particle  at  /' 
is  the  same  as  if  the  mass  of  the  shell  were  condensed  into 
its  centre. 

It  follows  from  this  that  a  sphere  which  is  cither  homo- 
geneous or  consists  of  concentric  spherical  shells  of  uniform 
density,  attracts  tiie  i)iiiti(le  at  P  in  the  same  manner  as  if 
the  whole  mass  were  collected  at  its  centre. 

(2)  Let  the  particle,  /'.  be  inside  the  sphere.  Then  we 
proceed  exactly  as  lieforc.  and  obtain  eijuation  (2).  which  is 
true  whether  the  particle  be  outside  or  inside  the  sphere  • 


228  EXAMPLES. 

but  the  r-limits  in  tliis  case  are  a  —  c  and  a  -\-  c.     Hence 
from  (a)  we  have,  by  i)erforming  tlie  ^-integration, 

attracMon  of  sl.ell  =  ^-/^"(l  -  ''^)dr. 


therefore  a  particle  within  the  siiell  is  equally  attracted  in 
every  direction,  i.  o.,  is  not  attracted  at  all. 

OoR. — If  a  particle  be  inside  a  hoinogoiious  sphere  at  the 
distance  r  from  its  centre,  all  that  porjion  of  the  sphere 
which  is  at  a  greater  distance  from  the  centre  tlu'n  the 
particle  i)roduces  no  effect  on  the  particle,  wliile  the  re- 
mainder of  the  sphere  attracts  the  particle  in  the  same 
manner  as  if  tin-  mass  of  the  remainder  were  all  collected 
at  the  centre  of  the  sphere.  Thus  the  attraction  of  tlie 
sphere  on  the  particle 

_  l^rpr^  4Trpr 

—       ■;        or         - — • 
H  3 

Heiue,  within  a  homogeneous  sphere  the  attraction  varies 
Its  the  distance  from  the  centre. 

The  propositions  respecting  the  attraction  of  a  uniform 
spherical  shell  on  an  external  or  internal  particle  were 
gi  en  by  Newton  (Principia,  liib.  I,  Prop.  70,  71).  (See 
Todhunter's  Statics,  p.  275,  also  Pratt's  Mechs.,  p.  137. 
Price's  Anal.  Mechs.,  Vol.  I,  p.  2G6,  Minchin's  Statics, 
p.  403). 

EXAMPLES. 

1.  The  span  AB  —  800  feet,  and  CO  =  IfiOO  fc<'t,  find 
tlie   length   of  the  curve,  C.i.  the    hcigiit.   (1{,  and    the 


-f  c.    Hence 
ition, 


y  attracteil  in 

sphere  at  the 
of  the  splieie 
lire  Ihi'n  the 
while  the  re- 
in the  same 
1  all  collected 
iiction  of  the 


acfion  varies 

of  a  uniform 
)article  were 
\  71).  (See 
3hs.,  p.  137. 
lin's  Statics, 


000  feet,  find 
7/,  and   the 


EXAMPLES.  229 

inclination,  B,  of  the  curve  to  the  horizon  at  either  point  of 
suspension. 

(1)  Here  ^_  =  {,  and  e  =  2-71838, 


therefore 


and 


e«  =  (2.71828)*  =  1-2840, 
e~«  =  (2-71828)"*  =  0-7788. 


Suhstituting  these  values  in  (5)  we  get 

S  =  800  X  0-5052  =  404-16. 
CA  =  404-16  feet. 


Hence 
(2) 


(8) 


therefore 


=  800  X  2-0628  —  1600 
=  50- 24  feet 

tan  e  =  f^  =  i(e*  -  e-i),  from  (1), 

=  0-2526, 
e  =  14°  11'. 


s  404-16 

Otherwise  tan  8  =  -,  from  (a),  =  -^jt^/t"  =  0-2526,  aa 

hcfore. 


1600 


2.  The  entire  load  on  the  cord  in  Fig.  71  is  160000  lbs., 
(he  span  is  192  ft.,  and  the  height  is  ITi  ft.;  find  the  tension 
at  the  points  of  8U|)port,  and  also  Ihe  tension  at  the  lowest 
uoint.  Ans.  Tension  at  one  end  =  268208  lbs. 

Horizon  (a!  teiie»on  =  256000  '• 


^m 


230 


EX  AM  PL  KS. 


3.  A  chain,  AOB,  10  feet  long,  and  weighing  GO  lbs.,  is 
snspcndcd  so  (hat  the  lieight,  67/,  —  4  feet  ;  find  the 
horizontal  tension,  and  the  inclination,  0,  ot  the  chain  to 
the  horizon  at  the  points  of  su[)p()i't. 

A)is.  Horizontal  tension  =  3|  lbs.,  6  =  77°  19'. 

4.  A  chain  110  ft.  long  is  suspended  from  two  points  in 
the  same  horizontal  plane,  108  ft.  apart ;  show  that  the 
tensi(m  at  the  lowest  point  is  1.477  times  the  weight  of  the 
chain  nearly. 


hing  ZO  lbs.,  is 
X'ct ;  find  the 
t  the  chain  to 

'  =  77°  19'. 

two  j)oiiits  in 
liow  that  the 
weight  of  the 


PART    II. 


KINEMATICS   (MOTION). 


CHAPTER    I. 

RECTILINEAR    MOTION. 

134.  Definitions. —Velocity. —Kinematics  is  that 
hranch  of  Dynamii-s  which  tieatn  of  motion  without  refer- 
ence to  the  bodies  moved  or  the  forces  producing  tlie  mo- 
tion (Art.  1).  Altliough  we  do  not  know  motion  as  free 
from  force  or  from  tlie  mc'Jer  that  is  moved,  yet  there  are 
cases  in  wiiich  it  is  advantageous  to  separate  the  ideas  of 
force,  matter,  and  motion,  and  to  study  motion  in  the 
abstract,  /.  e.,  without  any  reference  to  tohat  is  nwviii;/,  or 
the  cause  of  motion.  'I'o  tlie  study  of  pure  motion,  then, 
we  devote  this  and  the  following  chapter. 

The  velocity  of  a  particle  has  been  defined  to  be  its  rntc 
ofviofioii  (Art.  0).  The  formula'  for  uniform  and  variable 
velocities  are  those  which  were  deduced  in  Art.  7.  From 
(1)  antl  {i)  of  that  Art.  we  have 


V  = 


V  = 


ds 

ifr 


(1) 


in  which  r  is  the  velocity,  .s  the  space,  and  t  the  time. 


232 


EXAMPLES. 


J  EXAMPLES. 

1.  A  body  moves  at  the  rate  of  754  yards  per  hour.    Find 
the  velocity  in  feet  per  second. 
Since  tlie  velocity  is  uniform  we  use  (1),  hence 


J 


s       754  X  3 
V  =  -  =  ~ —  =  0.628  ft.  per  sec,  Atis. 


t        00  X  60 


2.  Find  the  position  of  a  particle  at  a  given  time,  t, 
when  the  velocity  xaries  as  the  distance  from  a  given  point 
on  the  rectilniear  i)ath. 

Here  the  velocity  being  variable  wo  have  from  (2) 


ds       , 


where  k  ia  a  constant; 
(Is 


therefore 


=  kdi]    .'.    \og8  =  kt  +  c. 


(1) 


where  c  is  an  arbitrary  constant. 

Now  if  wo  suppose  that  s^  is  the  distance  of  the  particle 
from  the  given  point  when  t  —  Q  we  have  c  =  log  «„, 
which  in  (1)  gives 

log  —  =z  kt;    or    s  =  s^e**. 
«o 

^  3.  A  railway  train  travels  at  the  rate  of  40  miles  per 
liour  ;  find  its  velocity  in  feet  per  second. 

Ans.  58.06  ft.  per  second. 

V  4.  A  train  takes  7  Ii.  31  ni.  to  travel  200  miles  ;  find  its 
velocity.  Ans.  39.02  ft.  per  sec. 

\j  5.  U  s  =  4/'^  lind  the  velocity  at  I  he  end  of  five  seconds. 

Ans.  300  ft.  per  sec. 

^6.  Find  the  position  of  the  particle  in  Ex.  2,  when  the 
velocity  varies  as  the  time.  Ans.  s  =  s^  +  ^kt\ 


r  hour.    Find 

ticc 

sec,  Ans. 

ven   time,  t, 
given  point 

tn(2) 


(1) 


the  particle 
c  =  log  So, 


to  miles  por 

er  second. 

les  ;  find  ita 
t.  por  sec. 

five  seconds. 
t.  pel-  sec. 

2,  when  the 


1. 1  /  >  ^7o 

I  0  *  cU. 


/  0 


!  0-A 


V 


ACCELKRATTOX  ZERO. 


iC.  l[ 
233 


^7.  Find  the  distance  the  particle  will  move  in  one 
minute,  when  the  velocity  is  10  ft.  at  the  end  of  one 
second  and  varies  as  the  time.  Ans.  18000  ft. 

135.  Acceleration. — Acceleration  has  been  defined  to 
be  the  rats  of  change  of  velocity  (Art.  8).  It  is  a  velocity 
increment.  The  formulae  for  acceleration  are  from  (1),  (2), 
and  (3)  of  (Art.  9), 


f- 


t 


1 

f      dv 

J-Tt' 

\:' 

i) 

.       d^s 

] 

J  —  ITi* 

(1) 

(2) 
(3) 


(1)   being  for  uniform,  and    (2)    and    (3)    for  variable, 

acceleration/ 

If  the  velocity  decreases,  f  is  negative,  and  (2)  and  (3) 

become 

do  f     ^  —       f' 

I '     ,112  —  ~  / » 


dt 


df^ 


and  the  velocity  and  time  are  inverse  functions  of  each 
other. 

136.  The  Relation  between  the  Space  and  Time 
when  the  Acceleration  =  0. 


Hero  we  have 


dt^ 


=  0, 


80  that  if  v^  is  the  constant  velocity  we  have 

da 
dl='^\ 

.«,     8  =  v„t  4-  «o» 


* 


234 


ACCELh'RATTOX  CONSTANT. 


in  wliich  .<„  is  the  space  wiiicli  tlio  body  has  pabsed  ovor 
when  /  =  0.  If  /  is  computed  from  the  time  the  body 
starts  from  rest,  then  s  =  v^t.  The  student  will  observe 
tiial  this  is  a  ease  of  uniform  velocity. 

137.  The  Relation  (1)  between  the  Space  and 
Time,  and  ('i)  between  the  Space  and  Velosity, 
w'aen  the  Acceleration    s  Constant 


(I)  Let  A  be  the  initial  position  of     o         a         p       ' 
the   particle   supposed    to    be    moving  ^'8-'' 

toward  the  right,  P  its  position  at  any  time,  /,  from  A,  v 
its  velocity  at  that  time,  and  /the  ccnstant  acceleration  of 
its  vf'locity.  Take  any  fixed  point,  0,  in  the  line  of  motion 
as  o'-igin,  and  let  OA  =  Sg  ;  OP  =  s.  Then  the  equation 
of  aioti(>u  is 


as 
dt 


=  ft-\-c. 


Suppose  the  velocity  of  Mie  particle,  at  the  jmint  A  to  bo 
r„,  then  when  ^  =  0,  v  =  r^;*  hence  c  =  i'^,  and 


.-.    3  =  \fl^  +  rj  +  c. 
But  when  f  =  0,  s  =  x^;  iience  c'  =  Sg,  and 
s  ^  |.//«  +  r,/  +  .So, 


CO 


(■^) 


Heii.'e  if  a  ])articl('  nK>ins  from  r.'.-^t  f''  "n  ilie  oi  i^in  O,  wiih 
•!  ciin.qant  acccli-nilion.  \\(    iiave 


•  Called  inilial  velocity  and  upacc  roHpcctlvoly,  or  tho  velocity  Ibc  ptrtlcle  has, 
aiiu  i<pnc«  It  bati  lauvcd  uvvr  at  tho  iiistau.  t  bc^iim  to  be  reckoucd. 


aiMtimm 


LS  passed  over 

me   the  body 

will  observe 


Space  and 
I  Velocity, 


Fig. 73 

t,  from  A,  V 
celcratioii  of 
ne  of  motion 
the  equation 


(1) 


nt  A  to  bo 
and 

(3) 


(3) 

,.in  f>,  will, 

;hc  particle  haa, 


A^CELEJiAr/OX    VARIAULE. 
S  =  \ft\ 


235 
(4) 


«nd  thus  the  space  described  varies  as  the  square  of  the 
time. 


(2)  From  (1)  we  have 


dp 
dJ 


'ifs  +  0. 


But  when  s  =  s^,  v  =  r,,;  lionco  C  =  v^'  —  2/Sg,  and 

therefore 

i^  =  2fs  -f-  r„2  _  2/s„.  (5) 

Ecjiiations  (2)  and  (3)  give  the  velocity  and  position  of  the 
particle  in  terms  of  / ;  and  (5)  gives  the  velocity  in  terms 
of  s. 

138.  When  the  Acceleration  VarieB  directly  as 
the  Time  from  a  State  of  Rest,  find  the  Velocity 
and  Space  at  the  end  of  the  Time  t 


Ilere 


dt^ 

ds 
dt 


=  at; 


where  I'o  is  the  initial  velocity  ; 

the  initial  spax^c  being  0  since  I  is  estimated  from  rest. 

139.  When  the  Acceleration  Varies  directly  as 
the  Distance  from  a  given  Point  in  the  line  of  Mo- 
tion, and  is  negative,  find  the  Relation  between 
the  Space  and  Time. 


^'>ti  EXAMPLES. 

by  calling  «„  the  >ttift.  of  *■  when  the  piirticle  is  at  rest. 

••• --^r^^  =  k^dt, 

the  negative  sign  being  taKo./  since  the  particle  is  moving 
towards  the  origin  ; 

.'.    cos-'—  ^  kk, 
if  « "=  Sg  when  /  =  0 ; 

.••    «  =  So  co«*M<. 


EXAMPLES 

1.  A  body  commences  to  move  with  «  \clocity  of  30  ft. 
per  sec,  and  its  velocity  is  increased  m  ouch  .second  by 
10  ft.     Kind  the  si)ace  described  in  5  seconds. 

Here  f  =z  10,  f„  =  30,  a„  =:  0,  and  .  =.  o,  therefore 
from  (;{)  we  have 


v) 


s-  =  i  •  10  .  25  +  30  .  5  =  275,  Ans. 


2.  A  body  starting  with  a  velocity  of  10  ft.  per  )thc  ,  and 
moving  with  a  constant  ijcceleration,  descrilws  90  ft.  in 
4. sees. ;  find  tlie  aeeolenitioii.  Ans.  (!'  ft,  per  .see. 

'.'!.  Kind  the  velocity  of  a  body  which  starting  from  rest 
with  an  acceleration  of  10  ft.  per  sec,  has  descril)ed  a  space 
"^20  ft.  ^„^,.  20  ft. 


FALLING    BODIES. 


231 


at  rest. 


)  18  moving 


y  of  30  ft. 
.second  by 

i,  tlicreforo 


!T  sfcc  ,  and 
i  90  n.  in 
.  per  siH!. 

:  from  nsl 
:)ed  a  space 
lA'.  20  ft. 


4.  Througli  what  space  must  a  liody  pass  under  an  accel- 
eration of  5  ft.  per  sec,  so  that  its  velocity  may  iucrcade 
from  10  ft.  to  20  ft.  per  sec.  ?  Ans.  30  ft. 

N^.  In  what  time  will  a  Ixxly  moving*  with  an  accelera- 
tion of  25  ft.  per  sec,  acquire  a  velocity  of  1000  ft.  per 
second?  '!««•  40 '^^'^'S- 

'^  fi.  A  body  starting  from  rest  has  been  moving  *or  5  min- 
utes, and  has  accjuired  a  velocity  of  'M)  miles  an  hour; 
what  is  the  acceleration  iu  feet  per  second  ? 

Ans.  {I  ft.  per  sec. 


/ 


n.  If  a  body  moves  from  rest  with  an  acceleration  of  |  ft. 
per  sec,  liow  long  must  it  move  to  ac(iuire  a  velocity  of 
40m:!esanhour?  ^1"^.  88  sees. 

140.  Equations  of  Motion  for  Falling  Bodies.— 

Tiie  most  important  case  of  the  motion  of  a  particle  with  a 
constant  acceleration  in  its  line  of  motion  is  that  of  a  body 
moving  under  the  action  nf  gravity,  which  for  small  dis- 
tanc-'es  above  the  earth's  surface  may  be  considered  constant. 
When  a  body  is  allowed  to  fall  freely,  it  is  found  to  acquire 
a  velocity  of  about  'M.2  feet  per  second  during  every  second 
of  Its  motion,  so  that  it  moves  with  an  acceleration  of  32.2 
feet  per  second  (Art.  21).  This  acceleration  is  less  at  tne 
summit  of  a  high  mountain  than  near  the  surface  of  the 
earth  ;  and  less  at  the  etiuator  than  in  the  neighborhood  of 
the  poles ;  i.  e.,  the  velocity  which  a  body  actpiires  in  falling 
freely  for  one  second  varies  with  the  lalihide  of  the  place, 
and  with  its  altitude  above  the  sea  level  ;  but  is  independ- 
ent of  the  size  of  the  body  and  of  its  mass.  Practically, 
however,  bodies  do  not  fall  freely,  as  the  resistance  of  the 
air  ojiposes  their  motion,  and  therefore  in  practical  ciises  at 
high  speed  (c  f/.,  in  artillery)  the  resistance  of  the  air  must 
be  taken   into  account.     But  at  present  we  shall  neglect 

♦  In  each  case  the  body  is  Dupposed  t"  marl  from  reel  uiilesB  othorwiBc  stated 


238 


FALLr.\G  no  ores. 


this  resistance,  and  consider  the  hodies  as  moving  in  vacuo 
under  the  action  of  gravity,  t.  e.,  with  a  constant  accelera- 
tion of  about  '.I'i.'i  feet  per  second. 

As  neitlier  iho  substance  of  the  body  nor  tlie  cause  ot 
the  motion  needs  to  be  taken  into  consideration,  all  prol)- 
lems  relating  to  falling  hodies  may  be  regarded  as  eases  of 
accelerated  motion,  and  treated  from  purely  gcometru  * 
considerations.  Therefore  if  we  denote  the  acceleration  by 
g,  as  in  Art.  2',),  and  consider  the  particle  in  Art.  137  to  be 
moving  vertically  downwards,  then  ('i),  (3),  (5)  of  Art.  137 
become,  by  substituting^  foi'/j 


V  =  gt  +  ?•„, 

s  =  W^  +  ''oi  +  •%» 
v^  =  2gs  +  j'o'  —  '^gs^, 


(A) 

,1    '•■I  .> 


s  being  measured  as  before  from  a  fixed  point,  0,  in  the 
line  of  motion. 

Suppose  the  ]>article  to  be  projected  downward  from  O, 
then  A  commences  with  O  and  .v^  =  0.  Hence  (A)  be- 
comes 

V  =  gt  +  Vo,  (1) 


v^  =  2gs  +  Vo'. 


(2) 
(3) 


As  a  particular  case  suppose  the  particle  to  be  dro])ped 
from  rest  at  0  (Fig.  73).  Then  A  coincides  with  O,  and 
.Sj  =  0,  r„  =  0.     llei   e  equations  (A)  become 


V  =  gt, 

s  =  yt^, 

1)2  =  igs. 


(4) 
(5) 
(6) 


I 


ing  in  vacuo 
ant  accele ra- 
the cause  of 
oji.  all  prol)- 
(1  as  cases  of 
y  geonietru  * 
^deration  Ijy 
rt.  137  to  be 
)  of  Art.  137 


(A) 

.  i .-(  v.A ,  > : 

t,  0,  in  the 

ird  from  O, 
3ncc  (A)  be- 

(1) 
(2) 
(3) 

be  dropped 
vith  O,  and 


(4) 
(5) 
(6) 


i 


PARTiri,E   VliOJECTED    UPWARDS. 


239 


141.  When  the  Particle  is  Projected  Vertically 
Upwards. — Here  if  we  measure  .s'  upwards  from  the  point 
of  projection,  0,  the  acceleration  tends  to  diminish  tlic 
space  and  therefore  tiie  acceleration  is  negative,  and  the 
equation  of  motion  is  (Art.  135) 


(Ps 


=  —II' 


In  other  respects  the   solution    is   the  same 
therefore  *• 
obtain 


Taking 


J  =  0  in  (A)  and  changing  the  sign  of  (j,*  we 

'"  =  I'o  —  'Jt>  (1 ) 

8  =  vJ.-  \fjP,  (2) 

v*  =  v,^  -  2ffs.  (3) 

CoR.  1.  —  77/c  time  darimj  ivhich  a  j)article  rises  when 
projected  vertically  upwards. 

When  the  particle  reaches  its  highest  point,  its  velocity 
is  zero.  If  therefore  we  put  r  =  0  in  (1),  tiie  correspond- 
ing value  of  t  will  be  the  time  of  the  particle  ascending  to  a 
state  of  rest. 

ff 

Cor.  2. — The  time  of  fliyht  before  returning  to  the  start- 
ing point. 

From  (i)  we  have  the  distance  of  the  particle  from  the 
starting  point  after  /  seconds,  when  projected  vertically 
upwards  with  the  velocity  v^.  Now  when  the  particle  has 
risen  to  its  maximum  height  and  returned  to  the  point  of 
])roje('tion.  .v  =  0.  If.  tiierefore.  we  put  .v  =  (t  in  ('I),  and 
solve  for  /,  we  shall  get  the  time  of  flight.     Therefore, 


*  g  '\<i   positive  or   negative  accordinj; 
ceuding. 


as  the  particle  is  deBcending  or  as- 


wnm 


^iO  i'AKTIVLh    rilO.IEVTKD    ITWAliDS. 


which  gives 


2?' 
t  =  0,     or     -^. 
9 


The  first  value  of  t  shows  the  time  before  tlie  particle 
starts,  the  latter  shows   the   time  wh''n  it  has  returned. 

Hence,  the  whole  lime  of  lliglit  is  — ",  which  's  just  double 

the  time  of  rising  ((.'or.  1) ;  that  \&,the  time  of  rising  equals 
the  time  of  falling. 

The  final  velocity,  by  (!)  of  Art.  140,  z=  gt  z=z  g  x^-^ 

(C!or.  1)  zjz  r„  ;  hcr.co  a  body  returns  to  any  \m\\i  in  its 
path  with  the  sai.ie  velocity  at  which  it  left  it.  In  other 
words,  a  body  passes  each  point  in  its  path  with  the  same 
velocity,  wholher  rising  or  falling,  since  the  velocity  at  any 
point  may  be  considered  as  a  velocity  of  projection. 

Cor.  3. — The  greatest  hcigut  to  which  the  particle  will 
rise- 

At  the  summit  v  =  (»,  and  the  c.n-respondiii<i  value  of  s 
will  be  the  greatest  heigiit  to  which  the  particle  will  rise  ; 
when  V  =  0,  (3)  becomes 

fo^  =  ''igs ; 


.'.     s  =    i- 


Von.  4. — Since  ?'„»  —  'igs,  where  s  is  the  height  from 
whicii  u  liody  fails  to  gain  I  he  velocity  ''„•  ''  follows  thai  ii 
body  will  rise  through  the  same  space  in  losing  a  velocity 
?'„  as  it  would  fall  thn  ugh  to  gain  it. 


«MM 


tlie  particle 
s  returned. 

just  double 

ising  equals 

(  =  f/  y~~ 

point  in  its 
.  In  other 
li  the  same 
)oity  at  any 
ion. 

mrtide  will 

value  of  s 
e  will  rise  ; 


ei}j;lit  from 
lows  thai  a 
;  a  velocity 


EXAMVLES. 


EXAMPLES, 


241 


L 


^  1.  A  body  projected  vertically  downwards  with  a  velocity 
of  20  ft.  a  see.  from  the  top  of  a  tower,  reaches  the  ground 
in  3.5  sees.-,  tind  the  height  of  the  tower. 

Here  t  =  2^,   and   r„  -^  20  ;    assume  g  =  32.     Then 
from  (2)  of  Art.  140  we  have 

g  —  iB^sjL  4-  20  X  I  =  150  ft. 

^'  2.  A  body  is  projected  vertically  upwards  with  a  velocity 
of  200  ft.  per  second ;  find  the  velocity  with  which  it  will 
pass  a  point  100  ft.  above  the  point  of  projection. 

Here  v^  =  200,  .s  =  100  ;  therefore  from  (3)  we  have 
1)2  =  40000  —  (5400  =  33(300 ; 
.  • .     V  =  40  a/21. 

'  3.  A  man  is  ascending  in  a  balloon  with  a  uniform 
velocity  of  20  ft.  per  sec,  wlieu  he  drops  a,  stone  whieli 
reaches  the  ground  in  4  sees.;  find  the  height  of  the 
balloon. 

Here  t'„  =  20,  and  /  ==  4  ;  therefore  from  (2)  we  have, 
after  clumging  the  sign  of  the  se(!on(l  menil)er  to  make  the 

result  positive, 

'  «  =  -  (80  -  25(i)  ==  170, 

which  was  the  height  of  tlie  balloon. 

^4.  A  body  is  projected  upwards  with  a  velocity  of  80  ft.; 
after  what  time  will  it  return  to  the  hand  ? 

.'1//.S'.  5  seconds. 

'^5.  With  what  velocity  must  a  body  be  projected   ver- 
tically uinvards  that  it  may  rise  40  ft.  ? 

Ann.  1(5  V 10  ft.  per  sec. 

11 


I* 


24;i  coMrosiTiny  of  vklocf'^ies. 

0.  A  l)()(ly  projected  vertically  ui)wards  passes  a  certain 
point  with  a  velocity  of  80  ft.  per  sec;  how  much  higher 
will  it  ascend  ?  Ans.  100  ft. 

T.  Two  balls  are  diop])od  from  the  top  of  a  tower,  one  of 
them  3  sees,  before  the  other  ;  how  far  will  they  l)e  apart 
5  sees,  after  the  tirst  was  let  fall  ?  Ans.  33fj  ft. 

V  8.  If  a  body  after  having  fallen  for  -'3  sees,  breaks  a  pane 
/)f  glass  and  thereby  loses  one-third  of  its  velocity,  find  the 
entire  space  through  which  it  will  have  fallen  iu  4  sees. 

.'1ms.  221  ft. 

142.  Composition  of  Velocities.— (1)  From  the  Par- 
nil  vtng  ram  of  Velan ill's,  (Art.  2!),  l-'ig.  2),  we  see  that  if  A  I? 
rcpreseiils  in  magnitude  and  direction  the  space  which 
would  l)e  described  in  one  second  by  a  particle  moving  with 
»  given  velocity,  and  AC  represents  in  magnitude  and 
(liit'cfion  the  space  which  would  be  descril)ed  in  one  second 
i)y  another  particle  moving  with  its  velocity,  then  Af).  //ir 
dimjonal  of  the  paralkloi/rom,  rcpreaenls  the  rvsuUant 
velocity  1)1  infit/uifitdr  and  direction, 

(2)  Hence  t/ie  resultant  of  any  two  velocities,  as  AH.  HI), 
(Kig.  2),  is  a  velocity  represented  by  the  third  side,  I).\.  «/' 
the  Irianyle  AHD;  and  if  a  point  hare  siniiillanronsly, 
velocities  repi-esented  by  A15,  HC,  CA.  the  sides  of  a  trian- 
<)le,  tahvn  in  the  samr  order,  it  is  at  rest. 

The  lines  which  arc  taken  to  rci)rescnt  any  given  forces 
may  clearly  be  taken  to  represent  the  velocities  which 
measure  tlu'sc  forces  (Art.  19),  therefore  from  the  Polyyon 
ami  I'arallelopipetl  of  Forces  the  Polyyon  and  Parallel- 
(ij)iped  of  Velocities  follow. 

(.'>)  Hence,  if  any  iiiimbrr  if  velorilies  tw  represnited  in 
viaynitiide  and  direction  t/y  /he  sides  if  a  closed  polyyon, 
taken  all  in  the  same  order,  ihe  resultant  is  zero. 

(4)  Also,  if  three  velocities  Ijr  reprc'^cnted  in  magnitude 


es  ii  certain 
ueh  liighor 
IS.  100  ft. 

>\vor,  one  of 
\v  he  apart 

3;j(]  rt. 

oaks  a  pane 
ty,  find  the 
1  4  sees. 
s.  -iU  ft. 

)m  tlie  /'ar- 

tliat  if  AH 

•ace  wliicli 

noviti<r  with 

riiitiide  and 

one  seeond 

len  AD.  //ir 

c    res  II II a  II  f 

IS  Ah,  ]]\), 
idc,  DA,  of 

ulliuiiniislii, 
of  a  Iriini- 

fiven  forces 
Hies  which 
he  Polijijon 
'I  Pam'llvl. 

ri'snitvil  ill 

"ll    IXlljljIDII, 

mar/ H  if  mil' 


f 


RESOLUTION  OF   VELOCfTrKS. 


243 


nnil  direction  by  the  three  edijes  of  a  paralleloptped,  the  re- 
sultant velocity  wilt  be  represented  by  tlic  diivjonat. 

(."))  When  there  are  two  velocities  or  three  velocities  in 
two  or  in  three  rectangular  directions,  tlie  resultant  is  the 
siiuare    root    of    the    sum    of    tl'.eir    squares.       Thus,    if 

ds    itx    dii    dz  .         1     •.•        ..  ^1  ■    ^       1 

-J,    .  ,    :  ,    y,  are  the  velocities  ot  the  inovnig  point  and 

its  components  parallel   to  the  axes,  we   have  from  {'Z)  of 
Art.  30, 


and  from  (1)  of  Art.  34, 


(2) 


143.  Resolution  of  Velocities.— As  the  diagonal  of 
tiie  parallelogram  (Fig.  2),  whose  sides  represent  the  com- 
ponent velocities  was  found  to  represent  the  resultant 
velocity,  so-any  velocity,  represented  by  a  given  straight 
line,  may  be  resolved  into  component  velocities  represented 
l)y  the  sides  of  the  parallelogram  of  which  the  gi\en  line 
is  the  diagonal. 

It  will  be  easily  seen  that  {'l)  of  Art.  134  is  ecjually 
applicable  whether  the  point  be  considered  as  moving  in  u 
straight  line  or  in  a  curved  line  ;  l)iit  since  in  the  latter 
case  the  direction  of  motion  continually  changes,  the  mere 
ivnoiiiit  of  the  velocity  is  not  sntticient  to  describe  the 
motion  completely,  so  it  will  l>e  necessary  to  know  at  every 
instant  the  direction,  as  well  as  the  nini/iiitinte.  of  tlu'  point's 
velocity.  In  such  cases  as  this  the  meliiod  commonly  em- 
ployed, whether  we  deal  with  velocities  or  accelerations, 
consists  mainly  in  studying,  not  the  velocity  or  acceleration, 
directly,  but  its  components  parallel  to  any  three  assumed 
rectangular  axes.     If  t  lie  particle  beat  the  point  (.r,  y,  z), 


244 


EXAMPLES. 


at  tlie   time  I,  and   if  we   denote   its   velocities    parallel 
respectively  to  the  three  axes  by  u,  v,  w,  we  have 


ax 
lit 


=  v^; 


ily 

lit 


(Iz 


-"   dt~  ^'' 


Denotinfi:  by  v  the  velocity  of  the  moving  ])article  along 
the  curve  at  tlie  time  /,  wo  Inive  as  above 


da  n<ix^      [di/\^      ldz\^ 

and  if  «,  (i,  y  be  the  angles  wiiich  the  direction  of  motion 
along  the  curve  makes  witli  the  axes,  we  have,  as  in  {t)  of 
(Art.  34), 

(Ix        (^  ^ 

-T-  =  —  cos  «  =  V  COS  «  =  Vx ; 
dt       dt 

dy       ds        ^  _ 

dz        ds 

-J-  =  -J-,  cos  y  =  v  cos  y  =  v.. 
dl        dt 

'Ix    cltJ    (iz 
Hence  each   of  the  components   --.r,    j',  -r,  is  to  be 

found  from  the  whole  velocity  by  resolving  tlie  velocity, 
i.  v.,  l)y  multiplying  tiie  velocity  by  conine  of  the  anyle 
between  the  direction  of  motion  and  that  of  the  compo- 
nent. 

EXAMPLES. 

'^1.  A  body  moves  under  the  indnence  of  two  velocities, 
at  I'ight  anglcH  to  ciU'li  dtju'r,  ('(|na!  respectively  to  17.14  ft„ 
Mild  i;{.  11  ft.  per  second.  l-'ind  the  magnitude  of  tlie 
resultant  motion,  and  Ihe  angles  into  wliicli  it  divides  the 
right  angle. 

Ann.  -H.bV.i  ft.  per  see. ;  '.iT  25'  and  52"  35'. 


MOTIOS   OS     l.V   IM'L/.\Kli   PLASE. 


'i\ti 


c'S    parallel 


tide  along 


(1) 

of  motion 
8  in  ('I)  of 


,  is   to   be 

he  velocity, 
'  the  tvif/lc 
tlie  conipo- 


I  velocities, 

0  17.Uft. 
ide  of  the 
divides  tiie 

1  55}"  35'. 


^2.  A  ship  sails  due  north  at  the  rate  of  4  knots  p-s*- 
hoHF,  and  a  ball  is  rolled  towards  the  east,  across  her  deck, 
at  right   angles   to   her  motion   at   the  rate  of  10  ft.  per 
second.     »'ind  the  magnitude  aiul  direction   of  the   real 
motion  of  the  ball. 

Ans.   12.07  ft.  per  see.;  and  N.  50^  H. 

y  3.  A  boat  moves  N.  30°  E.,  at  the  rate  of  0  miles  jur 
hour.     Find  its  rate  of  motion  northerly  and  easterly. 

Ans.  5.2  miles  per  liour  north,  and  3  miles  per  hour 
east. 

144.  Motion  on  an  Inclined  Plane.— By  an  exten- 
sion of  the  eciuations  of  Art.  140,  we  may  treat  the  case  of 
a  particle  sliding  from  rest  down  a  smooth  inclined  ])lane. 
As  this  is  a  very  simple  case  in  whicli  an  acceleration  is 
resolved,  it  is  convenient  to  treat  of  it  in  this  part  of  our 
work  ;  yet  as  it  properly  belongs  to  the  theory  of  con- 
strained motion,  we  are  unable  to  give  a  comi)lete  solution 
of  it,  until  the  principles  of  such  motion  have  been  ex- 
plained in  a  future  chapter. 

Let  P  be  the  position  of  the  particle  at 
any  time,  /,  on  the  inclined  plime  OA,  OP 
=  .«,  its  distance  from  a  fixed  point,  0,  in 
the  line  of  motion,  and  let  «  be  the  inclina- 
tion of  OA  to  the  horizontal  line  AH.  Let 
Pi  re[)resent  //,  the  vertical  acceleration  with  '"'B-'* 

which  the  body  would  move  if  free  to  fall.  Resolve  this 
into  two  components,  V<t  =  </  sin  «  along,  and  P^  =  ,r/ 
cos  «  perpendicular  to  OA.  The  comiiou-ut  //  cos  a  pro- 
duces pressure  on  the  plane,  but  does  not  affect  tlie  motion. 
The  oidy  acceleration  down  the  iilane  is  that  component  of 
the  whole  acceleration  whicii  is  parallel  to  the  plane,  viz., 
g  sin  «.     The  equation  of  motion,  therefore,  is 


dt^ 


=  (J  sin  «, 


(1) 


Aim 


24G  DESCENT  DOWN  CHORDS   OF  A    CIRCLE. 

the  solution  of  which,  as  g  sin  «  is  constant,  is  included  in 
that  of  Art.  140;  and  all  the  results  for  particles  moving 
vertically  as  given  in  Arts.  140  and  141  will  he  made  to 
apply  to  (1)  hy  writing.'/  sin  «  for//.  Thus,  if  the  particle 
he  projected  down  or  up  the  pkne,  we  get  from  (1),  (a),  (;5) 
of  Arts.  140  and  141,  by  this  means 

V  =  t'o  ±6' sin  «•/,  (2) 

s  z=z  v^t  ±  y  sin  «•  f\  (3) 

i^  =  i'o^  ±  2</  sin  «•.?,  (4) 

in  which  the  -f  or  —  sign  is  to  be  taken  according  as  the 

body  is  projected  duion  or  np  the  plane. 

If  the  particle  starts  from  rest  from  0,  we  get  from  (4), 

(5),  (G)  of  Art.  140 

i;  =  (/  sin  a-i,  (5) 

s  =  iff  sin  «•  t\  (6) 

V*  =  2ff  sin  «.,s.  (7) 

Cor.  1. — The  velocity  acquired  'by  a  particle  in  falliiu, 
(loivu  a  given  inclined  plane. 

Draw  PC  parallel  to  AB  (Fig.  74),  then  if  v  be  the 
velocity  at  P,  we  have  from  (7) 

v'  =  2(/  sin  tfS 
=  2g-0a 

Hence,  from  (G)  of  Art.  140  the  velocity  is  the  same  at  P 
as  if  the  particle  had  fallen  througli  tlie  vertica!  space  OC  ; 
tiiat  is,  the  velocity  acquired  in  falling  dunm  a  smooth 
inclined  plane  is  the  same  as  would  be  acquired  in  falling 
freely  t/irough  the  perpendicular  height  of  the  plane. 


t 


DESCENT  DOnW   CUOIIDS   OF  A    VIHVLE. 


(2) 

(3) 

(4) 
ng  as  the 

from  (4), 

(5) 

(6) 

(7) 


t 


Cor.  2. —  When  the  particle  in  projected  up  the  plane  luith 
a  given  i  hcity,  to  find  how  high  it  will  ascend,  and  the  time 
of  ascent. 

From  (4)  we  have 


v^  =  vj'  —  2g  sin  «•«. 


When  V  =  0  the  particle  will  stop ;  hence,  the  distance  it 
will  ascend  will  be  given  by  the  equation 

0  =  I'o^  —  2g  sin  ««s, 


s  = 


Hg  sin  « 

To  find  the  time  we  have  from  (2) 

V  =  Vq  —  g  sin  n't; 
and  the  particle  stops  when  i*  =  0,  in  which  case  we  have 

t  =  -^^. 
g  sin  a 

From  (6)  we  derive   the  following  curious  and  useful 
result. 

145.  The  Times  of  Descent  down 
all  Chords  drawn  from  the  Highest 
Point  of  a  Vertical  Circle  are  equal.— 

Ix't  AB  be  the  vertical  diameter  of  the 
circle,  AC  any  cord  through  A,  «e  its 
inclination  to  the  horizon  ;  join  HC  ;  then 
if  /  be  the  time  of  descent  down  AC  wc 
have  from  (0)  of  Art.  144 


Bat 


AC  =  yt^  sin  «. 
AC  =  AH  sin  a; 


Fig.75 


^aM 


248 


Lh\E   ■)F  QI'IVKEST  DESCBNT. 


.-.     Ml  =  ^gt\ 


or 


t 


=  V  y 


vliich  13  constant,  and  shows  that  the  time  of  falling  down 
u  \\  t'luml  is  the  same  as  the  time  of  falling  down  the 
diameter. 

"IJoK.— Similarly  it  may  be  shown  that  the  times  of 
defeoent  down  all  cliords  drawn  to  B,  the  lowest  point, 
are  equal  ;  tiiat  is.  tlie  time  down  C15  is  oqiial  to  hat 
down  AB. 

146.  The  Straight  Line  of  Quickest  Descent  from 
(1)  a  Given  Point  to  a  Given  Straight  Line  (2)  from 
a  Given  Point  to  a  Given  Curve. 

(1)  Let  A  be  the  given  point  and  BC 
the  given  line.  Through  A  draw  the 
horizontal  line  AC,  meeting  CB  in  C; 
bisect  the  angle  ACB  by  CO  whicli  intcr- 
Kccls  in  0  tiie  vertical  line  drawn  through 
A  ;  from  0  draw  OP  perjiendicular  to  BC; 
join  AP  ;  AP  is  the  required  lino  of  quick- 
est de;iCC!lt. 

For  OP  is  evidently  equal  to  OA,  and  therefore  the 
circle  described  with  O  as  centre  and  with  OP  (=  OA)  for 
radius,  will  touch  the  line  BO  at  P,  and  since  the  time  of 
falling  down  all  chords  of  this  circle  from  A  is  the  same, 
AP  must  be  the  line  of  quickest  descent. 

(2)  To  find  the  straight  line  of  quickest  descent  to  a 
given  curi'p,  all  that  is  required  is  to  draw  a  circle  having 
the  given  point  as  the  upper  extremity  of  its  vertical 
diameter,  and  tnngent  tj  the  curve.  Hence  if  DE  (Fig. 
70)  be  the  curve,  A  the  point,  draw  AH  vertical;  aad,  with 
centre  in  AH,  describe  a  circle  passing  tlirough    A,  and 


piling  down 
down   tiie 


le  times  of 
)\vost  point, 
(iial   to    liat 


scent  from. 
Qe  (2)  from 


lerefore  tlie 
(=  OA)  for 
the  time  of 
is  the  same, 

?scent  to  a 
rclo  hiiviiitr 
its  vertical 

f   DE   (Fi<r. 

;  aad,  with 
gh    A,  and 


SXASiPhBS. 


240 


couching  DE  iit  P,  then  Al   .s  the  requirecl  line.     For,  if  we 

take  any  other  point,  Q,  in  DE,  and  draw  AQ  cutting  tlic 

circle  in  q,  then  liic   time  down  AP  --—  time  down  Ay< 

•m"     ,.vvn  AQ.     Hence  AP  is  the  line  of  (piickest  descent. 

The  proh'-'u.  -T  ''nding  the  line  of  (juickcst  descent  from  ii  point  to 
a  line  or  curve  is  thus  "lund  to  resolvi'  itself  into  the  [uirt  ly  geometric 
problem  of  drawing  a  circle,  tlie  biphest  ]M)int  of  wliich  sliall  be  the 
given  point  and  which  shall  touch  the  given  line  or  curve. 

EXAMPLES.* 

1.  If  the  earth  travels  in  its  orl)it  600  million  mile'-  r-.x 
3()5J  days,  with  uniform  motion,  what  is  its  veloci'  'u 
miles  per  second  ?  •  Atis.   19-01  mi- '«. 

•^  2.  A  train  of  cars  moving  with  a  velocity  of  20  m.jos  ^" 
hour,  had  been  gone  3  'lours  when  a  locomoi  e  i\,.;' 
dispatched  in  pursuit,  with  a  velocity  of  25  miles  -  'on  ; 
in  what  time  did  the  latter  overtake  the  former  ? 

,  Ans.  12  hours. 

J 
3.  A  body  moving  from  rest  with  a  uniform  acceleration 

describes  90  ft.  in  the  5th  second  of  its  motion  ;  find  the 

acceleration,/,  and  velocity,  v,  after  10  seconds. 

Am.  f  =z  %0;  V  —  200. 

■^  4.  Find  the  velocity  of  a  particle  which,  moving  with  an 
acceleration  of  20  ft.  per  sec.  has  traversed  1000  ft. 

A  ns.  200  ft.  per  sec. 

5.  A  body  is  observed  to  move  over  45  ft.  and  55  ft.  in 
two  successive  seconds  ;  find  the  space  it  would  describe  in 
the  20th  second.  Ans.  195  ft. 

G.  The  velocity  of  a  body  increases  every  hour  at  tiie  rate 
of  360  yards  per  hour.  What  is  the  acceleration,/,  in  feet 
per  second,  and  what  is  the  space,  .s,  describeil  from  rest  in 
20  seconds?  .l//.s.  /  =  0.3;  s  —  60  ft. 

♦  In  these  exannik^H  take  g     aa  ft. 


ioO 


EXAMPLES. 


V  7.  A  body  is  moviiifr.  at  :\  given  instant,  at  the  rate  of 
8  ft.  per  sir.;  at  tlio  end  of  5  sees,  its  vciofity  i.-'  I'J  ft.  per 
sec.  Assnniing  its  acceleration  to  l)e  utiifonn,  what  was  its 
velocity  at  tlic  end  of  4  sees.,  and  wliat  will  be  its  velocity 
at  the  end  oflO  sees.  ?  Ans.   lU-8;  30. 

"^8.  A  body  is  moving  at  a  given  instant  with  a  velocity  of 
30  miles  an  bonr.  and  comes  to  rest  in  11  sees.;  if  the 
retardation  is  uniform  what  was  its  velocity  5  sees,  before  it 
stopped  ?  Ans.  20  ft.  per  sec. 

^  9.  A  body  moves  at  the  rate  of  12  ft.  a  sec.  with  a 
nniform  acceleration  of  4  ;  (1)  state  exactly  what  is  meant 
by  the  number  4  ;  (2)  suj)pose  the  acceleration  to  go  on  for 
5  sees.,  and  then  to  cease,  what  distance  will  the  body 
describe  between  the  ends  of  the  5th  and  12th  sees.? 

Ans.  224  ft. 

10.  A  body,  whose  velocity  undergoes  a  uniform  retardii- 
tion  of  8,  describes  in  2  sees,  a  distance  of  30  ft.;  (1)  what 
was  its  initial  velocity  ^  (2)  How  much  longer  than  the 
3  sees,  would  it  move  before  coming  to  rest  ? 

AH.<i.  (1)  23;  (2)  I  sec. 

11.  A  body  whose  motion  is  uniforndy  retarded,  changes 
its  velocity  from  24  to  (1  while  describing  a  distance  of  12 
ft.;  in  what  time  does  it  descnbe  the  12  ft.? 

Ans.  0-8  sec. 

12.  The  velocity  of  a  bmly,  which  is  at  first  G  ft.  a  sec, 
undergoes  a  uniform  acceleration  of  3  ;  at  the  end  of  4  sees, 
the  acceleration  ceases  ;  how  far  does  the  body  move  in  10 
sees,  from  the  beginning  of  the  motion?        Ans.  15G  ft. 

13.  A  body  moves  for  a  quai-ter  of  an  hour  with  a  uni- 
form acceleration  ;  in  the  first  5  minutes  it  describes  3r»0 
yards;  in  the  second  5  minutes  420  yards;  what  is  the 
whole  distance  describtid  in  a  quarter  of  an  hour? 

Ans.   1200  vds. 


EXAMPLES. 


•251 


tlie  rate  of 
i.-'  I'J  ft.  ])er 
AvAi  was  its 
its  velocity 
U-8;  30. 

I  velocity  of 
3C.S.;  if  the 
cs.  before  it 
.  per  sec. 

sec.  with  a 
it  is  meant 

0  go  on  for 

1  the  body 
?cs.  ? 

\  224  ft, 

m  retardiu 
.;  (1)  what 
if  than  the 

;2)  I  sec. 

h1,  changes 
tance  of  12 

0-8  sec. 

G  ft.  a  .sec, 
d  of  4  sees. 
nove  in  10 
.  156  ft. 

I'itli  a  iiiii- 

scribes  3r>0 

hat  is  the 
p 

200  vds. 


14.  Two  sees,  after  a  body  is  let  fall  another  body  i^ 
|)roje'jted  vertically  downwards  with  a  velocity  of  Kto  It. 
lier  sec;  when  will  it  overtake  (he  former':' 

A)is.    1 J  sees. 

15.  A  body  is  projected  npwanls  with  a  velocity  of  KtO 
ft  per  sec;  lind  the  whole  time  of  llight. 


.•I//.S'.  0]  sees 


'  It!.  A  balloon  is  rising  uniformly  with  a  velocity  of  10  ft. 
per  sec,  when  a  man  drops  from  it  a  stone  which  reaclu'S 
the  ground  in  3  sees.;  tind  the  height  of  the  balloon,  (1) 
when  the  stone  was  dropped;  and  (2)  when  it  reached  the 
ground.  A„x.   (1)  114  ft.;  (2)  144  ft. 

17.  A  man  is  standing  on  a  platform  which  descends 
with  a  uniform  acceleration  of  5ft  per.«ec. ;  after  havinjj 
descended  for  2  sees,  he  drops  a  ball  ;  what  will  be  the 
velocay  of  the  ball  after  2  more  seconds?         Ann.  74  ft. 

'  18.  A  ballojn  has  been  ascending  vertically  at  a  uniform 
rate  for  4- 5  sees.,  and  a  stone  let  fall  from  it  reaches  tlie 
ground  in  7  sees.;  find  the  velocity,  r,  of  the  balloon  and 
the  height,  s,  from  which  the  stone  is  let  fall 

Ans.  V  =  174f  ft  per  sec;  s  =  784  ft.  If  the  ballo.)u 
is  still  ascending  w'lon  the  stone  is  let  fall  v  =  G8-17  ft. 
per  sec;  .s  =  300.76  ft.? 

Kll.  With  what  velocity  must  a  particle  be  projected 
downwards,  that  it  may  in  /  s'cs  overtake  another  jjarticle 
which  has  alreadv  fallen  through  i  "i.  'i 


/ 


Anf^-  "  ==  I  +  V2r///. 


20.  A  person  while  ascending  in  a  balli.oi?  with  a  vertical 
velocity  of  F  ft.  per  sec,  lets  fall  a  stone  wlien  ho  is  //  fi- 
ai)ovo  the  ground;  reciuired  the  time  in  which  t!'C  stou- 


will  roach  the  g.'ound. 


Ans. 


V  +  V  V^  +  ty/h 

7 


252 


EXAMPLES. 


21.  A  body,  A,  is  i)n)jected  vortical!}'  downwards  from 
tlie  top  of  a  tower  witli  tlie  velocity  V,  and  one  sec.  after- 
wards aiiotlier  Itody,  H,  is  let  fall  from  a  window  n  ft.  from 
tiio  toj)  of  the  tower  ;  in  what  time,  /,  will  A  overtake  B  ? 

22,  A  stone  let  fall  into  a  well,  is  heard  to  strike  the 
bottom  in  t  seconds  ;  reijuired  the  depth  of  the  well,  sup- 
posing the  velocity  of  sound  to  bo  a  ft.  per  sec. 


Anx. 


V 


.  I*  It 


23.  A  stone  is  dropped  into  a  well,  and  aUer  .3  sees,  the 
sound  of  the  splash  is  heard.  Find  the  depth  to  the 
surface  of  the  water,  the  velocity  of  sound  being  1127  ft. 
per  sec.  A>i>i.   132.9. 

•^  24.  A  body  is  simultaneously  impressed  with  three 
uniform  velocities,  one  of  which  would  cause  it  to  move 
10  ft.  North  in  2  sees.;  another  12  ft.  in  one  sec.  in  the 
same  direction;  and  a  third  21  ft.  South  in  3  sees.  Where 
will  the  body  be  in  5  sees.  ?  Ans.  50  ft.  North. 

25.  A  boat  is  rowed  across  a  river  1{  miles  wide,  in  a 
direction  making  an  angle  of  87°  with  the  bank.  The 
boat  travels  at  the  rate  of  5  miles  an  hour,  and  the  river 
runs  at  the  rate  of  2.3  miles  an  hour.  Find  at  what  point 
of  the  opposita  <mk  the  boat  will  land,  if  the  angle  of  87° 
be  made  against  thj  stream. 

Ans.  898  yards  down  the  stream  from  the  opposite 
point. 

26.  A  body  moves  with  a  velocity  of  10  ft.  per  sec.  in  a 
given  direction  ;  find  the  velocity  in  a  direction  inclined  at 
an  angle  of  30°  to  the  original  direction. 

Ans.  5  Vo  ft.  per  sec. 


^M 


nards  from 
sec.  after- 
n  It.  from 

rtakc  B  ? 

"±1_ 

strike  the 
B  well,  sup- 

.3  sees,  the 
pth  to  the 
iig  Wilit. 

s.  r.i-i.9. 

with  three 
it  to  move 
sec.  in  the 
scs.  Where 
t.  North. 

wide,  in  a 
auk.  The 
d  the  river 
what  point 
ngle  of  87° 

le  opposite 

5r  sec.  in  a 
inclined  at 

per  sec. 


EX  A  MPLES. 


253 


JJ7.  A  smooth  plane  is  inclined  at  an  angle  of  30°  to  the 
horizon  ;  a  body  is  started  up  the  plane  with  the  velocity 
b(j ;  find  when  it  is  distant  9//  from  the  starting  point. 

Ans.  2,  or  18  sees. 

■^8.  Tiie  angle  (»f  a  plane  is  30°;  find  the  velocity  with 
which  u  l)ody  must  be  projected  np  it  to  reach  the  top, 
the  length  of  the  plane  being  ^0  ft. 

Am.  8  VlO  ft.  [)er  sec. 

'^W.  A  body  is  projected  down  a  plane,  the  inclination  of 
which  is  45°,  with  a  velocity  of  10  ft.;  find  the  space 
described  in  2|  sees.  Ans.  95.7  ft.  nearly. 

''  30.  A  steam-engine  starts  on  a  downward  incline  of 
1  in  200*  with  a  velocity  of  7^  miles  an  honr  neglecting 
friction  ;  find  the  space  traversed  in  two  minnt 

.I//.V.   824  yards. 

''  31.  A  body  projected  up  an  incline  of  1  in  100  with  a 
velocity  of  15  miles  an  hour  just  reaches  the  summit ;  find 
the  time  occupied.  Ans.  68.75  sees. 

■^  32.  From  a  point  in  an  inclined  plane  a  body  is  made  to 
slide  up  the  ))lane  with  a  velocity  of  Ki.l  ft.  per  sec.  (1) 
How  far  will  it  go  before  it  comes  to  rest,  the  inclination 
of  the  plane  to  the  horizon  being  30°  ?  (2)  Also  how  far 
will  the  body  be  from  the  starting  point  after  5  sees,  from 
the  beginning  of  motion  ? 

Ans.   (1)  8.05  ft. ;  (2)  120.75  ft.  lower  down. 

v/  ... 

33.  The  inclination  of  a  plane  is  3  vertical  to  4  hori- 
zontal ;  a  body  is  made  to  slide  up  the  incline  with  an 
initial  velocity  of  30  ft.  a  sec;  (1)  how  far  will  it  go  before 
beginning  to  return,  and  (2)  after  how  many  seconds  will 
it  r    urn  to  its  starting  point? 

Alls.   (1)  33J  ft.:  (2)  3J  sees. 


*  All  iiirliiie  of  1  in  800  mean?  hero  1  foot  vortically  to  a  Ungth  of  200  ft.,  thoug)) 
it  is  used  b    En(;inccr«  to  mean  1  foot  vertically  to  aiO  ft.  horizontaUy. 


254  EXAMPLES. 

34.  There  is  an  inclined  iiiuno  of  5  vortical  to  1:^  hori- 
zontal, a  bod^  .slides  down  5:>  ft.  of  its  length,  and  then 
l)as.ses  without  kiss  of  veiocity  on  to  the  horizontal  jdane; 
after  how  long  from  the  beginning  of  the  motion  will  it  be 
at  a  distance  of  100  ft.  from  the  foot  of  the  incline  y 

Ans.  5.7  sees. 


/ 

as. 


A  body  is  projected  up  an  inclined  i)]ane,  whose 
length  is  10  times  its  height,  with  a  velocity  of  .'JO  ft.  per 
sec. ;  in  wliat  time  will  its  velocity  be  destroyed  ? 

Ans.  ^  sees.,  iff/  —  32. 

30.  A  body  falls  from  rest  down  a  given  inclined  plane; 
compare  the  times  of  describing  the  first  and  last  halves 
*^^  ''^-  Ans.  As  1  :  V^  -  1. 

37.  Two  bodies,  projected  down  two  planes  inclined  to 
the  horizon  at  angles  of  45°  and  (10°,  descrU)e  in  the  same 
time  spaces  respectively  as  V'i  :  V3  ;  find  the  ratio  of  the 
niitial  velocities  of  the  projected  bodies. 

Ans.   -y/a  :  \/;\. 

38.  Through  what  chord  of  a  circle  must  a  body  fall  to 
acquire  half  the  velocity  gained  by  falling  through  the 
diameter? 

Ans.  The  chord  which  is  inclined  at  00°  to  the  vertical. 

/  30.  Find  the  velocity  with  which  a  body  should  be  ))ro- 
jected  down  an  inclined  plane.  /,  so  that  the  time  of 
running  down  the  jjlano  shall  be  equal  to  the  time  of 
falling  down  the  height,  h. 

(I —  h  m\  «\ 

^  40.  Find  the  inclination  of  this  ]ii,;ne.  when  a  velocity 
(if  Jths  that  due  to  the  height  is  sullit'ient  to  render  the 
times  of  running  down  the  ]>lane,  and  of  falling  down  tho 
height,  ecjual  to  each  other.  Am,  30°, 


EXAMPLES. 


255 


to  1:^  hori- 
I,  and  thou 
iital  })liiiio; 
1  will  it  tit' 
110  y 
5. 7  sees. 

uie,  whosi' 
■  ;}0  ft.  per 

(J  ~  32. 

nod  pltiiie; 
last  lialves 
v/2  -  1. 

inclined  to 
I  the  same 
itio  of  the 

a :  V;j. 

ody  fall  to 
irough  tliG 

vortical. 

Id  be  ))vo- 
e  time  of 
10  time  of 

a  veloeity 
render  the 

down  the 
UK,  30°, 


il.  Through  what  eliord   of  a  eircle,  draAvn  from  the 
iKiltoiu  of  the  vertical  diameter  must  a  body  descend,  so  as 

to   afi|uin^  a  veloeity  equal   lo    -tli    part   of   the  velocity 

iu<iu!red  in  falling  down  tlie  vertical  diameter? 

Alts.  If  0  denote  the  angle  between  the  required  chord 

and  the  vertical  diameter  cos  6 


1 

n 


''•43.  Find  the  inclination,  d,  '.>f  the  radius  of  a  eircle  to 
the  vertical,  .«nch  that  a  body  running  down  will  c'escribe 
the  radius  in  the  san;o  time  that  anoMier  botly  requires  to 
full  down  the  vertical  diameter.  Am.   0  =  60". 

''43.  Find  the  inclination,  Q,  to  tl:o  vertical  of  tiie  diam- 
eter down  which  a  body  falling  will  describe  the  last  haif 
in  the  same  time  as  the  verticid  diameter. 

3  \/2  —  4 


Ans.  cos  6  = 


2\/5i 


41.  Show  that  the  times  of  descent  down  all  the  radii  of 

curvature  of  the  cycloid  (Fig.  40,  tJalculus)  are  equal;  that 

/Sr" 
is,  the  time  down  1*Q  is  equal  to  the  time  down  O'A  :=  y  --  • 

'45.  Find  the  inclination,  6,  to  the  horizon  of  an  inclined 
pliine.  so  that  the  time  of  descent  ..f  a  particle  down  the 
l.'iigth  may  be  «  times  that  down  the  height  of  the  plane. 

Ans.  6  =  sin^i  -  • 
n 

1(1.  I''ind  the  line  of  quickest  descent  from  the  focus  to 
a  paraltoia  whose  axis  is  vertical  and  vertex  upwards,  and 
show  that  its  length  is  equal  to  that  of  the  latus  rectum. 

17.  Find  the  line  of  quickest  descent  from  tiie  focus  oi  a 
\)arab()la  to  the  curve  when  Ihe  uxis  is  horizontal. 


256  EXAMPLES. 

'48.  Find  geometrically  the  line  of  quickest  descent  (1) 
from  a  point  within  a  circle  to  the  circle  ;  (2)  from  a  circle 
to  a  point  without  it. 

49.  Find  geometrically  the  straight  line  of  longest 
descent  from  a  circle  to  a  point  without  it,  and  which 
lies  below  the  circle. 

^'  50.  A  man  six  feet  high  walks  in  a  straight  line  at  the 
rate  of  four  miles  an  honr  away  from  a  street  hunp,  the 
height  of  which  is  10  feet;  supposing  the  man  to  start 
from  the  lamp-post,  find  the  j-ate  at  which  the  end  of  his 
shadow  travels,  and  also  the  rate  at  which  the  end  of  his 
shadow  separates  from  himself. 

Ans.  Shadow  travels  10  miles  an  hour,  and  gains  on 
himself  G  miles  an  hour. 

51.  Two  bodies  fall  in  tlie  same  time  from  two  given 
points  in  space  in  the  same  vertical  down  two  straight 
lines"  drawn  to  any  point  of  a  surface ;  show  that  the  sur- 
face is  an  equilateral  liyperboloid  of  revolution,  having  the 
given  points  as  \ertices. 

62.  Find  the  form  of  a  curve  in  a  vertical  i)lane,  such 
that  if  heavy  particles  be  simuUaneonsly  let  fall  from  each 
jioint  of  it  so  as  to  slide  freely  along  the  normal  at  that 
point,  they  may  all  reach  a  given  iiorizontal  straight  line  at 
tlio  same  instant. 

53.  Show  that  the  time  of  quickest  descent  down  a  focal 
chord  of  a  parabola  whose  axis  is  vertical  is 

V    g 

where  /  is  the  latup-  rectum. 

54.  Particles  slide  from  rcht  at  the  highest  point  of  a 
vertical  circle  down  chords,  and  are  then  allowed  to  move 


lesceni  (1) 
)m  a  circle 


of    longest 
and  wiiicli 


line  at  the 
;  lump,  the 
m  to  start 
end  of  his 
end  of  his 

.  gains  on 

two  given 
ro  straight 
it  the  sur- 
liaving  the 

)lane,  such 
from  eiicli 
lal  at  tiiat 
gilt  line  iit 

wn  a  focal 


point  of  a 
d  to  move 


EXAMPLES, 


857 


freely  ;  show  that  the  locus  of  the  foci  of  their  paths  is  a 
circle  of  half  the  radius,  and  that  all  the  paths  bisect  the 
vertical  radius. 

55.  If  the  particles  slide  down  chords  to  the  lowest  point, 
and  be  then  suffered  to  move  freely,  the  locus  of  the  foci  is 
a  cardioid. 

56.  Particles  fall  down  diameters  of  a  vertical  circle  ;  the 
locus  of  the  foci  of  their  subsequent  paths  is  the  circle. 


(I 


CHAPTER    II. 

CURVILINEAR     MOTION. 

147.  Remarks  on  Curvilinear  Motion.— The  mo- 
don,  whicli  wus  coiisidorod  in  tlie  last  elmptcr,  was  that  of 
■d  piU'ticle  describhig  a  rectilinear  i)at]i.  In  this  chapter  the 
cireunutanc'os  of  motion  in  which  the  ])atli  is  c»r/'j7t«ert/- 
will  he  considered.  The  concei)(ion  and  the  definition  of 
velocity  and  of  acfo'eration  which  were  given  in  Arts.  i;}4, 
135,  are  evidently  as  applicalile  to  a  particle  descrihing  a 
curvilinear  path  as  to  one  moving  along  a  straight  line; 
and  conseqnently  the  fornnila'  for  velocity  in  Arts.  142, 14:S, 
are  applicable  either  to  rectilinear  or  to  curvilinear  motion. 
In  the  last  chapter  the  effects  of  the  comi^sition  and  the 
re  .''ition  of  velocities  were  considered,  when  the  path 
ta^ven  by  the  particle  in  consequence  of  them  was  straight ; 
wo  have  now  to  investigate  the  effects  of  velocities  and  of 
accelerations  in  a  more  general  way. 

148.  Composition  of  Uniform  Velocity  and  Ac- 
celeration. "  Su|ipose  a  l)ody  tends  to  move  in  one  direc- 
tion witli  a  uniform  velocity  which  would  carry  it  from  A 
to  B  in  one  second,  and  also  subject  to  an 
acceleration  that  would  carry  it  from  A 
to  C  in  one  second ;  then  at  the  end  of 
the   second    the   body  will    be  at  ]),   the 
opposite  end  of  the  diagonal  of  (he  j)ar-  \  p.  „ 
allelograni  AMDC.  just  as  if  it  had  moved 
from  A  to  B  and  tlu'u  from  Biol)  in  the  second,  but  the 
l>ody  will  move  in   the  ritnr  ami   not  along  (he  i/iiii/oiuiL 
I'or,  the  body  in  its  motion  is  making  progress  uniformlv 
in  the  direction  AH,  at  (be  sanit^  rale  us  if  it  had  no  other 
motion;  and  at  the  sumo  time  it  is  being  uccelcrated  in  the 


—The  mo- 
ras  thut  of 
cliaptiT  tlic 
rttrriiinear 
cfiiiitioii  of 

Arts.  i;U, 
['scrihiug  a 
liglifc  line  ; 
S.M2,  ]4;i, 
ar  motion. 
n  and  tlio 

tlio  path 
s  straight ; 
:ics  and  of 

and  Ac- 

ono.  diroc- 
it  from  A 


Fig.?? 


I,  Imt  the 
tlidjliiiial, 
inifornily 
no  olhi'r 
od  iu  tho 


COMPOSITION  OF  ACCELERATIOyS. 


259 


direction  AC,  as  fast  as  if  it  had  no  other  motion.  Henco 
the  Lody  will  rcjich  D  a.s  far  from  the  line  AC  as  if  it  had 
moved  over  AB,  au^  as  far  from  AB  as  if  it  had  moved 
over  AC ;  but  since  tl.c  velocity  along  AC  is  not  uniform, 
tho  spaces  described  in  (((ual  intervals  of  times  will  not  be 
equal  along  AC  while  th'jy  are  e(iual  along  AB.  and  tiicre- 
fore  the  points  «,,  a^,  a^,  will  not  bo  in  a  straight  line.  In 
this  case,  therefore,  the  i)ath  is  a  curve. 

149.  Composition  and  Resolution  of  Accelera- 
tions.— If  a  body  is  subject  to  two  dififerent  accelerations 
in  different  directions  the  sides  of  a  parallelogratn  may  Ijo 
taken  to  represent  the  Component  Accelerations,  aiul 
tho  diagonal  will  rejjresent  the  liesnltant  Accclerafion, 
although  the  i)ath  of  the  body  may  bo  along  some  other 
line. 

Rem. — Tliese  results  with  those  of  Arts.  14^,  14.'5,  may  be 
summed  up  in  one  general  law:  When  n  body  tends  to 
■inove  with  several  different  velocities  in  different  directions, 
the  tmly  will  be,  at  the  end  of  any  yiven  time,  at  the  same 
point,  (IS  if  it  had  moved  with  each  velocity  separately. 
This  is  the  fundamental  law  of  Die  composition  of  veloci- 
ties, and  it  shows  that  all  pn)l)loms  which  involve  tendon-  • 
cios  to  motion  in  different  directions  simultaneously,  may 
be  treated  as  if  those  tendencies  were  successive.* 

(Ps 
If   ..-^  bo  the  acceleration  along  the  curve,  and  (a;,  y,  z) 

be  the  place  of  the  moving  particle  at  the  time,  t,  it  is 
evident  tiiat  the  component  accelerations  parallel  to  tho 
(P.r    (Py    dh 


axes  arc 


nave 


dt^ '  dp  '  dp 


(Px 
ilp 


Denoting  these  by  «x,  tty,  us,  wc 
<Pz 


KX', 


(Py 
dp 


—  «« 


and  '\/uJ  -\-  k/  +  «««'  is  the  result  ant  arrelrration. 


*  Sou  Kumai'kH  ud  Ncwton'i*  td  law,  Art.  166. 


260  COMPOSITION  OF  ACCKLEnATIONS. 

Also  if  n,  (i,  y,  be  lliu  uiiglos  wliich  the  direction  o! 
motion  makes  with  the  axes,  we  have 

d^x       (Ps 

cPz       cPs 

^3  =  ^  COS  y  =  «.. 

d-1 
The  acceleration  ^^,  is  not  generally  the  complete  resultant  of  the 

three  component  acccleintions,  but  is  so  only  when  the  path  is  a 
Straight  iine  or  the  velocity  is  zero.     It  is,  however,  the  only  part 

of  tlieir  resultant  which  has  any  effect  on  the  velocity.     —  is  the 

sum  of  the  resolved  parts  of  the  coniiwnent  accelerations  in  the  direc- 
tion  of  motion,  aa  the  following  identical  equation  shows: 

^  _  dx  d-x      dy  d'y      dz   d'e 
dt^~  (h-  dfi  ■*■  ils  •  dt''  "•"  ds  ■  dT'' 

which  follows  immediately  from  (1)  of  Art.  143  by  differentiation. 
Accelerations  are  therefore  subject  to  the  same  laws  of  composition 
and  resolution  as  velocities  ;  and  cnsequently  the  acceleration  of  the 
particle  along  any  lino  is  the  sum  of  the  resolvid   partH  of  ilie  axial 

accelerations  along  that  line.  Tliusto  find  V'!,  the  acceleration  nloiur  » 
^^^  has  to  1)0  multiplied  by    -,  which   is  the  direction-cosine  of  tlie 

Bmull  iir<-  ilx.  'I'he  other  part  of  the  resultant  is  at  right  angles  to 
this,  and  its  only  effect  is  to  change  the  dircrthm  of  the  motion  of  the 
]ioint.  (See  Talt  and  Steele's  Dynamics  of  a  Particle,  also  Thomson 
and  Tait's  Nat  Phil.) 

The  following  arc  oxamplos  in  hIiIcIi  Iho  iircccdiiiij  cx- 
prcssioiiK  aiv  iii)i)lif(l  to  oases  in  which  the  laws  of  velocity 
and  of  acceleration  are  given. 


direction  o! 


ultant  of  the 

he  patli  IB  a 
he  only  part 

f.     ^,18  the 

1  in  the  direc- 

i: 


fforentiation. 

composition 

ration  of  the 

!  of  the  axial 

ition  iiloiig  s, 

'osino  of  ilio 

jht  nnplcs  to 
notion  of  the 
Ib')  T)ionisoi) 


ccdiiiy   ('\- 
of  velocity 


EXASIl'LES. 


EXAMPLES. 


261 


1.  A  particle  moves  so  that  the  axial  components  of  its 
velocity  vary  as  the  corresponding  co-ordinates ;  it  is 
re(iuired  to  find  the  equation  of  its  path ;  and  the  accel- 
erations along  the  axes. 

Here  |  =  A:a:;    |  =  %; 

...    ^  =  '^  =  kdt; 
X       y 

...    log?  =  !ogf  =  A.^ 

if  {a,  b)  is  the  initial  place  of  the  particle, 
...    X  =  a<M\    y  =  6e«- 


'  '    a      h 

is  the  equation  of  the  path. 
And  the  axial  accelerations  are 


m,. 


2.  A  wheel  rolls  along  a  straight  line  with  a  uniform 
velocity  ;  compare  the  velocity  of  a  given  iwint  in  the  cir- 
cumference with  that  of  the  centre  of  the  wheel. 

Lot  the  line  along  which  the  wheel  roils  be  the  axis  of  x, 
and  let  o  be  the  velocity  of  its  centre;  then  a  point  in  its 
circumference  describes  a  cycloid,  of  which,  the  origin 
being  taken  at  its  starting  point,  the  equation  is 


X 


=  a  vers-i  ^  —  (iay  —  .'/')*; 


263  EXAMPLES. 


But 


dx  _         (ly        _     da 
d  (         ^,y\  a  dy 


\  a' 


dt  \  af        ('},,„  „  ./2»i    dt 


ds  __  ds 
J/  ~  dy 


CZay  -  f)^ 


dt    -  \a)    '^' 


whicl)  is  tlio  velocity  of  the  point  in  the  circumference  of 
the  whci'l,  'I'hiis  the  velocity  of  the  highest  point  of  the 
tvheel  is  twice  as  great  as  that  of  the  centre,  while  the 
point  that  is  in  contact  with  the  straight  line  has  no 
velocity.     (See  Price's  Anni.  Mecfi's.,  Vol.  I,  p.  41 G.) 

dx  (In 

3.  ]i  -jj  =  ky,  -:-  =  kx,  show  that  the  path  is  an  equi- 
lateral hyperbola  and  that  the  axial  components  are 

(Px  _  <iPy  _ 

dfi  -  '^ ""'  dt^  -  '^y- 

4.  A  particle  describes  an  cllij)se  so  that  the  a;-component 
of  its  velocity  is  a  constant,  «  ;  find  the  ^-component  of  its 
velocity  and  acceleration,  and  the  time  of  describing  the 
ellipse. 

Let  the  equation  of  the  ellipse  bo 

rt^  ^    ft2   ~    ^  ' 

and  let  {x,  y)  be  the  position  of  the  particle  at  the  time  t  ; 

,,  dx  ,     dy  b'^x 

then  -,-  =  «  ;    and      /  = — ; 

dt  '  dx  a^y' 

dy  _  dy    dx  _       afjf>    x 
dt  ~  <fx'  dt  ~  ~  n^  '  y' 

which  is  the  y-component  of  the  velocity. 


m3t  >'« 


feronct 

of 

int  of 

the 

while 

the 

lie  has 

no 

G.) 

is  an  equi- 

ire 

omponent 
lent  of  its 
ribing  the 


le  time  t ; 


EXAM  PLC. 

dx 

dy 

=z 

KlI 

/ 

■^Tt 

— - 

d^jf 

> 

26.S 


Also 


iience  the  acceleration  parallel  to  the  axis  of  //  varies 
inversely  as  the  cube  of  the  ordinate  of  the  ellipse,  and  acts 
towards  tlie  axis  of  a*,  as  is  shown  by  the  negative  sign. 

The  time  of  passing  from  the  extremity  if  the  minor 
axis  to  that  of  the  ma,  axis  is  found  hy  dividing  a  by  «, 
the  constant   velocity   parallel    to   the   axis   of  x,   giving 

-,  and  the  time  of  describing  the  whole  ellipse  is  —  • 
«  " 

If  the  orbit  is  a  circle  h  =  a,  and  the  acceleration  par- 

allel  to  the  axis  of  y  is  —  ^—^• 

If  the  velocity  parallel  to  the  //-axis  is  constant  and  equal 

to  /3,  then 

dx  _  _a^(i    1/ , 


dt 
df 


fp     X 


Ab 


and  the  periodic  time  =  -^• 


x^      wS 
5.  A  particle  describes  the  hypc.bola  ^^ 


=  1  :  find 


w'       IP' 

(1)  the  acceleration  parallel  to  the  axis  of  x  if  the  velocity 
parallel  to  the  axis  of  y  is  a  constant,  3,  and  (2)  find  the 
acceleration  parallel  to  the  y-axis  if  the  velocity  paralN  *  '<> 
the  .r-axis  is  a  constant  <.:. 
(1)  Here  we  have 


dy 
dt 


0 ;    and 


dy  _  b^    x^ 
dx~  a^    y' 


EXAMPhES. 

dx  _  (l.v    (hj  _  /?«'    y 
•'■     M  ~  dy'  lit   ~'fJ^'x 

which  is  the  velocity  j)anillel  to  the  ^-axis. 

dy         dx 


Also 


dPx  _  fi(^ 
dfi  ~   ^  " 

_  ^ 


""dt-ydi 


hence  the  acceleration  parallel  to  the  x-axis  varies  inversely 
as  the  cube  of  tlie  abscissa,  and  the  a;-coniponent  of  the 
velocity  is  incrcsing. 


(2)  Here  we  have 


dx 
di 


.-=  a: 


and 


dy 

di 

^  _ 


«J2     X 


hence  the  acceleration  parallel  to  the  y-axis  is  negative  and 
the  ^-component  of  tiie  velocity  is  decreasing. 

G.  A  particle  describes  the  parabola,  x^  +  ^'  =  «*,  with 
a  constant  velocity,  c ;  find  the  accelerations  parallel  to  the 
fixes  of  X  and  y. 


Here  we  have 


ds 
di 


=  c; 


and 


dje  _  —  dy 


ds 


y 


*         {x  +  y)^' 


EXAMPLES. 


265 


and 


dr'       (Is^ 


<?x 


df^  ~  dt^   z  +  y       x  +  y' 
di^  ~  dP'  X  +  ~y       X  +  y' 


dififerentiating  we  get 


dfi 


jHay)i_. 

2{x  +  yf* 


ss  inversely 
lent  of  the 


dP  ~  2{.c  +  yf 

7.  A  particle  describes  a  parabola  with  such  a  varying 
velocity  that  its  projection  on  a  line  perpendicular  to  the 
axis  is  a  constant,  v.  Find  the  velocity  and  the  accelera- 
tion parallel  to  the  axis. 

Let  the  equation  of  the  parabola  be 

t/2  =  2px; 


then 


dt  -""' 


3gative  and 

=  «',  with 
allel  to  the 


and 


dx 
di 


dx    dy 
dy    dt 


which  is  the  velocity  parallel  to  x 


Also 


t 
P' 


which  shows  that  the  particle  is  moving  away  from  the 
tangent  to  the  curve  at  the  vertex  with  a  constant  accelera- 
tion. 


2W 


I'h'o.iErriLE  i.\  r.irro. 


Ili'iice  as  the  eartli  acts  on  jjarticlcs  near  its  surface  with 
a  constant  acceleration  in  vertical  lines,  if  a  particle  is 
])rojc<'t('(l  with  a  velocity,  r,  in  a  horizontal  line  it  will  move 
in  a  panilxiliu  path. 

150.  Motion  of  Projectiles  in  Vacuo. — If  a  i)article 
he  projected  in  a  direetion  ohli(|ne  to  the  horizon  it  is 
called  a  Projectile,  and  the  i)ath  wiiicii  il  describes  is  called 
its  Trajvclory.  The  case  which  we  shall  liere  consider  is 
that  of  a  particle  movino;  in  vacuo  under  the  action  of 
frravity;  so  that  the  problem  is  that  of  the  nio/ion  of  a 
pnijerUIi'  in  mciio  ;  and  hence,  as  gravity  does  not  affect 
its  horizontal  velocity,  it  resolves  it.self  into  the  purely 
kinematic  problcjn  of  a  particle  moving  so  that  its  hori- 
zontal acceleration  is  0  and  its  vertical  acceleration  is  the 
constant,  </,  (Art.  140). 

151.  The  Path  of  a 
Projectile  in  Vacuo  is  a 
Parabola. — Let  tiie  plane 
in  which  the  particle  is  pro- 
jected be  tiie  ])lane  of  .ry; 
let  tl'.e  axis  of  x  be  horizon- 
tal and  the  axis  of  y  vertical 
and  positive  upwards,  the 
origin  being  au  the  i)oint  of 
jtrojection ;    let    the   velocity 

of  projecti(m  =  r,  and  let  the  line  of  jn-ojection  be  inclined 
at  an  angle  «  to  the  axis  of  x,  so  that  ;•  cos  «,  and  r  .sin  « 
are  the  resolved  parts  of  the  velocity  of  projection  along  tlie 
axes  of  .(•  and  //.  It  is  evident  tliat  the  ]>article  will  con- 
tinue to  move  ii!  tlic  ])lane  of  xy,  as  it  is  |)rojected  in  it. 
and  is  su'  iect  to  no  force  whicli  would  tend  to  wiiiidraw 
it  from  thu.  plane. 

Let  {x,  y)  be  the  jtlace,  /'.  of  the  particle  at  the  time  / ; 
then  the  equations  of  motion  are 


S           D 

E 

f 

< 

A- 

\. 

/ 

M         C 

a 

L 

Fig. 

78 

surface  with 
a  jjarticlo  is 
0  it  will  move 


-If  a  particle 
lorizon  it  is 
rihcs  is  called 
IV  consider  is 
lie  action  of 
vintiun  nf  a 
OS  not  affect 
)  the  j)urely 
hat  its  hori- 
ratiou  is  the 


1  he  inclined 
,  and  ;■  sin  « 
on  aloiifi;  the 
icie  will  con- 
•jected  in  il, 
to  withdraw 

the  time  / ; 


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iV 


PROJKCT/hh'   l.\    VACUO.  267 

the  iioc'ilc'ration  being  nogativc  siiico  the  ^-component  of 

the  velocity  is  ilocreatsiiig. 

The  first  and  second  integrals  of  those   equations   will 

then    bo,   taking   the   limits   corresponding   to  ^  =  /  and 

/  =  0, 

dx  dy  .  .  ^.. 

^-  =  V  cos  « ;  ^~  =  V  siu  ci  —  gt;  (1) 

X  =z  V  cos  {(t;   1/  =  V  sin  uf  —  y/l\  (5J) 

E(|uation8  (1)  and  (2)  give  the  coordinates  of  the  particle 
and  its  velocity  parallel  to  either  axis  at  any  tinio,  f. 
Eliminating  I  between  eipiations  (2)  we  obtain 


f/^ 


y  =  X  tan  «  —  .^  / — r" 
''  Zv'  cos''  « 


(3) 


which  is  the  C((uation  of  the  trajectory,  and  shows  that  the 
particle  will  move  in  a  {jurabola. 

152.  The  Parameter ;  the  Range  li ;  the  Greatest 
Height  II \  Height  of  the  Directrix.— E(piai ion  (:j)  of 
Art.  151  nniy  be  written 

„       2r'^  sin  rt  cos  a               2)1^  ens'*  « 
x* X  = y, 


(I 


9 


or 


1^  sin'^  «A 


/         i»^  sill  «  cos  «V  "Zv'cos'nl         j**  sin'^  «A   ,, 


Hy  comparing    this   with    I  he   ecjiiation   of   a   paraboin 
roforrel  to  its  vertex  as  origin,  wo  find  lor 


the  ttbscisaa  of  the  vertex  = 


(''■*  sin  a  cos  « 

g 


(^) 


268 


PROJECTILE  IN    VACUO. 


the  ordinate  of  the  vertex  =: 


v^  sin^  « 


2^      ' 


(3) 


the  paramder  (latus  rectum)  = (4) 


And  by  transferring  the  origin  to  the  vertex  (1)  becomes 


,  2v-  cos*  a 

a^= — y 


(5) 


wiiicli  is  the  equation  of  a  parabola  with  its  axis  vertical 
and  tlie  vertex  tiie  higlicst  point  of  the  curve. 

Tlie  distance,  OB,  between  the  point  of  projection  and 
the  point  wliere  the  pntjectile  strikes  the  horizontal  plane 
is  called  the  liidige  on  tiie  horizontal  jtlune,  and  is  tlie 
value  of  X  when  y  —  0.  I'utting  ^  =  0  in  (3)  of  Art.  151 
and  solving  for  x,  we  get 


the  horizontal  range  R  =  OB 


?/■*  sin  2a 


(<o 


which  is  evident,  also,  goometrically,  as  OB  =  200;  tluit 
is,  tiie  range  is  etjual  to  twice  the  abscissa  of  the  vertex. 

It  follows  from  (0)  that  the  range  is  the  greatest,  for  a 
given  velocity  of  projection,  when  «  —  45 ',  in  which  ca.se 

the  range  =  —  • 
ff 

Also  it  appears  from  (fi)  that  the  range  is  the  same  when 
a  is  replaced  l»y  its  complement  :  that  is,  lor  the  same 
velocity  of  projection  the  range  is  the  same  for  two  ditTer- 
ent  angles  tliiit  are  eomplements  of  each  other.  Il"  «  =  45° 
the  two  angles  iu'eoine  identical,  and  the  range  is  a 
mu\ininin. 

CA  is  eidled  ihv  (/rra test  /iriii/it,  II.  of  ilie  ))rojectile,  and 

,.3 

is  given  by  (;})  which,  when  <«  =  4.-)°  becomes  j--  (7) 


cos^ « 


(3) 


(^) 


1)  becomes 

(5) 

axis  vertical 

rojectioii  and 
izoiital  plane 
,  and  is  the 
)  of  Art.  151 


=  20C;  tiiat 
e  vt'itox. 
reatest,  for  u 
which  case 


3  same  when 
•r  tlie  same 
p  two  diffor- 
If  «  =  45° 
range    is    a 

ojectile,  and 

r         (7) 


VELOVITY   OF  Till-:   I'UD.IhU'TI hE. 

TJie  height  of  the  dinxlrir  —  CD 


t'2  sin^  a 


.  ii^  cos'"'  «      v^ 


269 


(«) 


Hence  when  «  =  45°  the  focus  of  tlie  parabola  lies  ii\ 
the  horizontal  line  through  the  point  of  projection. 

153.  The  Velocity  of  the  Particle  at  any  Point  of 

its  Path.— Let  V  be  the  velocity  at  any  point  of  its  path, 

then        F^  =  (||)'+  (;^)',  or  by  (1)  of  Art.  151 

=  v^  cos^  «  +  (('2  sin2  «  —  2/'  sin  itgt  +  (ff^) 

To  acquire  this  velocity  in  falling  from  rest,  the  particle 
must  have  fallen  through  a  height  ^j-,  (G)  of  Art.  140,  or 
its  equal 

=  PS. 

Hence,  the  velocity  at  any  point.  P,  on  the  curve  is  that 
which  the  ])article  would  acquire  in  falling  freely  in  vacuo 
down  the  vertical  height  SP;  that  is,  in  falling  from  the 
directrix  to  the  curve ;  and  the  velocity  of  i)rojection  at  0 
is  that  which  the  particle  would  actiuire  in  falling  freely 
through  the  height  CD.  The  directrix  of  the  puralmla  is 
therefore  determined  by  the  velocity  of  projection,  and  is  iit 
a  vertical  distance  above  the  point  of  i)rojection  equal  to 
that  down  which  a  i)article  falling  would  have  the  velocity 
of  projection 

154.    The  Time  of  Flight,  T,  along  a  Hoizontal 

Plane.— Put  ^  =  O  in  (IJ)  of  Art.  IT)!,  and  .solvi.  for  r.  the 


i'H) 


TDfE   OF  FLKlirr  OF  PliOJFCTILB. 


1         t     \  ■  \         A      J  2i''  sin  «  COS  rt     ^,  ,  ,,     , 
values  of  which  arc  0  and lint  tlie  liorizon- 


tal  velocity  is  v  cos  «.     Jli-nce  ^/le  time  of  Jlii/hf  — 


'if  sill  r< 


wiiicli  varies  as  the  sine  of  the  inclination  to  the  axis  of 


155.  To  Find  the  Point  at  which  a  Projectile  will 
Strike  a  Given  Inclined  Plane  passing  through  the 
Point  of  Projection,  and  the  Time  of  Flight— Let  the 

inclined  plane  make  an  angle  )J  with  the  horizon;  it  is 
evident  that  we  have  only  to  eliminate  y  between  y  =  x  tan 
0  and  (3)  of  Art.  lal,  which  gives  for  the  abscissa  of  the 
point  where  the  projectile  meets  ihe  plane 


a;,  = 


2v^  cos  «  sin  («  —  (3) 


(J  cos  (i 


and  the  ordinate  is 


(1) 


_  'iv^  cos  a  tan  /3  sin  («  —  /3) 


Hence  the  time  of  flight 


rp  _      Xj       _  2v  sin  («  —  (3) 
V  cos  «  g  cos  /3 


(2) 


156.  The  Direction  of  Projection  which  gives  the 
Greatest  Range  on  a  Given  Plane.— The  range  on  the 
horizontal  piano  is 

?'*  sin  2a 


which  for  a  given  value  of  v  is  greatest  wlion  a  =  -  (Art. 

152). 
The  range  on  the  inclined  plane  =  ,r,  sec  (3 


_  2v^  cos  n  sin  («  —  ^3) 
""  g  cos*  /3 


(1) 


AXOLB   OF  KLhVATlOX   OF  I'lioJECTthF. 


\n 


the  horizon- 

_  'It'  sill  fi 

~        .'/ 
iL'  axis  of  ./•. 

}jectile  will 
through  the 
;ht.— Let  tile 
jrizon  ;  it  is 
en  y  =  X  tan 
bscissa  of  tiie 


(1) 


(2) 

b  gives  the 

•ange  ou  tlio 


K  =  7  (Art. 


(1) 


To  find  tlie  value  of  «  which  niakes  tliis  a  maximum,  wo 
must  equate  to  zero  its  derivative  with  respect  to  «,  wliich 
gives 

cos  (2«  -  i3)  =  0 ; 


and  hence 


(2) 


(3) 


which  is  the  angle  which  the  direction  of  projection  makes 
with  the  inclined  plane  when  tlie  range  is  a  maximum; 
that  is,  the  projection  bisects  the  angle  between  the 
inclined  plane  and  the  vertical. 

In  this  case  by  substituting  ill  (1)  the  values  of  «  and 
(a  _  |3)  as  given  in  (2)  and  (3)  and  reducing,  we  get 


the  greatest  range  = 


v2 


g{l  -h  sin  j3) 


(4) 


157.  The  angle  of  Elevation  so  that  the  Particle 
may  pass  through  a  Given  Point — From  Art.  152, 
there  are  two  directions  in  whicli  a  particle  may  be  pro- 
jected so  as  to  reach  a  given  point ;  and  they  are  equally 

inclined  to  the  direction  of  projection  («  =  jj- 

Let  the  given  point  lie  in  the  plane  which  makes  an 
angle  p  witl»  tlio  horizon,  and  suppose  its  abscissa  to  be  h  ; 
then  we  must  have  from  (1)  of  Art.  155 


2v^ 


cos  «  sin  («  —  0)  =  h. 


g  cos  (i 

If  a'  and  «"  be  the  two  values  of  «  which  satisfy  tliia 
etiudtion,  we  must  have 

cos  «'  sin  {n  —  (i)  =  cos  «"  sin  (a"  —  P) ', 


jjf-i      EQl'ATlO.X  or  TR.U ECTOR r,   SECOXD  METHOD. 


and  therefore 


«"  -  /3  =  ^  -  «', 


or 


«"-4(.:+»)=s(.j+fl) 


«, 


(1) 


But  eaeli  member  of  (1)  is  the  angle  between  one  of  the 
direetions  of  projection  and  the  direction  for  the  greatest 
range  [Art.  15G,  (2)].  Hence,  as  in  Art,  152,  the  two 
direetions  of  projection  wliieii  enable  the  particle  to  pass 
through  a  j)oint  in  a  given  plane  through  the  point  of  pro- 
jection, are  equally  inclined  to  the  direction  of  projection 
for  the  greatest  range  along  that  plane.  (See  Tait  and 
Steele's  Dynamics  of  a  Particle,  p.  89.) 

158.  Second  Method  of  Finding  the  Equation  of 
the  Tr^yectory. — By  a  somewhat  simpler  method  than 
that  of  Art.  151,  we  may  find  the  e(|uation  of  the  ])ath  of 
the  i)rojectile  as  the  resultant  of  a  uniform  velocity  and  an 
acceleration  (Art.  148). 

Take  the  direction  of  projection  (Fig.  78)  as  the  axis  of 
X,  and  the  vertical  downwards  from  the  point  of  projection 
as  the  u-Kis  of  ij.  Then  (Art.  149,  Rem.)  the  velocity,  v, 
due  to  the  ])rojection,  will  carry  the  i)article,  with  uniform 
motion,  parallel  to  the  axis  of  j;  while  at  the  same  time,  it 
is  carried  with  constant  acceleration,  (/,  parallel  to  the  axis 
of  y.  Hence  at  any  time,  /,  the  eraations  of  motion  along 
the  axes  of  a;  and  y  respectively  are 

X  =  vt, 

y  =  W- 

That  is,  if  the  particle  were  moving  with  the  velocity  v, 
alone,  it  would  in  the  time  /,  arrive  at  Q;  and  if  it  were 
then  to  move  with  the  vertical  acceleration  y  alone  it  would 
in  the  same  time  arrive  at  P\  therefore  if  the  velocity  v 


METHOD. 


(1) 

?en  one  of  the 
)!•  tlie  greatest 
3. 52,  the  two 
article  to  pass 
point  of  pro- 
of projection 
See  Tait  and 


Equation  of 

method  than 
of  the  i)atli  of 
locity  and  an 

as  the  axis  of 
of  projection 
le  velocity,  «>, 
(V'ith  uniform 
same  time,  it 
'1  to  tlie  axis 
motion  along 


16  velocity  v, 
nd  if  it  were 
one  it  would 
he  velocity  v 


UXAMI'LES. 


273 


and  the  acceleration  g  are  simuUaneous,  the  particle  will  i^ 
the  time  i  arrive  at  P  (Art.  140,  Rem). 
Eli'iiinatiug  /  we  have 

which  is  the  e({uation  of  a  parabola  referred  to  a  diameter 
and  the  tangent  at  its  vertex.  The  distance  of  the  origin 
from   the  directrix,   being   |^tii  of  the  coetticient  of  y,  is 


'^g 


,  as  ill  Art.  152,  (8). 


EXAMPLES. 


1.  From  the  top  of  a  tower  two  particles  are  projected  at 
angles  «  and  i3  to  the  horizon  with  the  same  velocity,  v,  and 
both  strike  the  horizontal  plane  passing  through  the  bot- 
tom of  the  tower  at  the  same  point;  find  the  height  of 
the  tower. 

Let  h  =:  the  height  of  the  tower;  v  =r  the  velocity  of 
projection ;  then  if  the  ])article8  are  projected  from  the 
edge  of  the  top  of  the  tower,  and  x  is  the  distance  from  the 
bottom  of  the  tower  to  the  point  where  they  strike  the 
horizontal  plane  we  have  trom  (3)  of  Art.  151 


—  ?i  =  x  tan  «  —  "^  (1  -f  tan'  «), 


gx^ 


h  =  x  tan  /3  -  l^j  (1  -h  tan'  (3), 


by  subtraction 


X  =: 


2v^ 


g  (tan  «  -|-  tan  (i) 
which  in  (1)  or  (2)  gives 


2t^  cos  a  cos  0  _ 
«/sin(«  +  (i)  ' 


,  2iP  cos  «  cos  (i  cos  («  -f-  j3) 

-       Tpin  («  -I-  Hjy 


(1) 

(2) 


m 


274 


VSLOVITY  OF  DlSCUAHdK   OF  SllFl.LiS. 


2.  Particles  jiro  projectod  with  a  given  velocity  in  all 
lines  in  a  vertical  plane  from  tiu>  point  0;  it  is  re((uireil  tn 
Ind  the  locus  of  tiieir  highest  points. 

Let  (x,  y)  be  the  highest  point;  then  iVoni  (•■J)  and  (;J)  of 
Art.  152,  we  have 


X  :— 


i^  sin  «  cos  a 


y  = 


!/ 
r'  sin^  « 


•^ff 


therefore      sin^  «  =  ~-^ ,  and  oos'^  «  =  f-j-- 


Adding 


v^ 


4/  +  a« 


iv^y 


which  is  the  equation  of  an  ellijtse,  whose  major  axis  =  -  ; 

V*  .  .  '. 

and  the  minor  axis  =  — ;  and  the  origin  is  at  .the  extremity 

of  the  minor  axis. 


3.  Find  the  angle  of  projection,  <«,  so  that  the  area  con- 
tained between  the  patli  of  the  projectile  and  the  hori- 
zontal line  may  be  a  maximum,  and  find  the  value  of  the 
maximum  area. 

H        1 
Ans.  n  =  60°  and  Max.  Area  =    ■  ,  (3)*. 

4.  Find  the  ratio  of  the  areas  Ai  and  A,  of  the  two 
parabolas  described  by  projectiles  whose  horizontal  ranges 
are  the  same,  and  the  angles   of  projection  are  therefore 


complements  of  each  other. 


Ans.  ~  =  tan'  «. 
Aj 


159.  Velocity  of  Discharge  of  Balls  and  Shel's 
from  the  Mouth  of  a  Gun. —As  the  result  of  niinieiMii.v 


LLS. 


■locity    ill   all 
s  ro(|uireiI  In 

(v>)  and  (;})  of 


jr  axis  :=  — : 
ff 

the  c'xtromitv 


the  area  con- 
,nd  tlie  liori- 
e  value  of  the 

2  of  the  two 
zontal  ranj^es 
are  therefore 


=  tan*  «, 

and  SheFs 

of  11  inner.  I11.V 


AMllLAU    VEI.iK'iry.  ■.>;.) 

experinieiits  made  at  Woolwieli,  the  followini,'  t'nrimil;i  w;,.- 
regarded  as  a  eorreet  expression  for  the  velocity  of  liiilN  ;iii(i 
siiells,  on  .|iiitting  the  gun,  and  fired  with  iiiodeiair 
charges  of  powder,  from  the  pieces  of  orilnance  coininonlv 
used  for  military  purposes: 

where  r  is  the  velocity  in  feet  ])er  second,  P  the  weight  of 
the  charge  of  powder,  and  IT  the  weight  of  the  hall. 

For  tile  investigation  of  the  path  of  a  projectile  in  the 
atmosphere,  see  Chap.  I  of  Kinetics. 

160.  Angular  Velocity,  and  Angular  Accelera- 
tion.— Hitherto  the  method  of  resolving  velocities  i'kI 
accelerations  along  two  rectangular  axes  has  l)een  employed. 
It  remains  for  iis  to  investigate  the  kinematics  of  a  particle 
descriljing  a  curvilinear  path,  from  another  point  of  view 
and  in  relation  to  another  system  of  reference.  Hefore  we 
consider  velocities  ami  accelerations  in  reference  to  a 
system  of  polar  co-ordinates,  it  is  necessary  to  enquire  into 
a  mode  of  measuring  the  (ingular  irlocifi/  of  a  particle. 

Anyulnr  Velocity  may  he  drfiicd  ax  the  rti/c  of  aiiyiildr 
nio/inn.  Thus  let  (/•.  f))  he  the  jjosition  of  thi  point  /'.  and 
sii])pose  that  t'.ie  radius  vector  has  revolved  uniformly 
tiirougii  the  angle  0  in  the  time  /,  then  denoting  the 
angular  velocity  hy  o>,  we  shall  have,  as  in  linear  velocity 
(Art.  7) 


(.)  =z 


e 
t' 


If  however  the  radius  vector  does  not 
revolve  uniformly  througli  the  angle  6 
we  may  always  regard  it  as  revolving 
uniformly  through  the  angle  dO  in  the 
infinitesiriial  of  time  di  ;  hence  we  shall 
have  as  the  proper  value  of  w, 


n^7i 


r 


276 


EXAMI'LKS. 


U)  =: 


(Id 

dt' 


(J) 


Hence,  whether  tlie  angular  velocity  be  nniform  or 
variable,  it  is  the  ratio  of  the  angle  described  by  the  radiiia 
vector  in  a  given  time  to  the  time  in  which  it  is  described; 
thus  tlie  increase  of  the  angle,  in  angular  velocity,  take 
the  place  of  the  increase  of  the  distance  from  a  fixed  })oint, 
in  linear  velocity,  (Art.  7). 

Anf/ular  Acrelern/ioii  is  tltc  rafc  of  incri-ase  of  angular 
velocity  :  it  is  a  velocity  increment,  and  is  measured  in  the 
same  way  as  liiirar  acrclernlion  (Art.  !t).  Thus,  whether 
the  angular  acceleration  is  uniform  or  varialjle,  it  may 
always  be  regarded  as  uniform  during  the  infinitesimal  of 
time  dt  in  which  time  the  increment  of  the  velocity  will  bo 
d(>).  Hence  denoting  the  angular  acceleration  at  any  time, 
/.  by  <l>,  we  have 


^  = 


dt 


IQf'"""'<i> 


dfi' 


(2) 


and  thus,  whether  the  increase  of  angular  velocity  is 
uniform  or  variable,  tiie  angular  acceleration  is  the  increase 
of  angular  velocity  in  a  unit  of  time. 

The  following  examples  are  illustrations  of  the  preceding 
mode  of  estimating  velocities  and  accelerations. 

EXAMPLES. 


1.  If  a  particle  is  placed  on  the  revolving  line  at  the 
distance  r  from  the  origin,  and  the  line  revolves  with  a 
uniform  angular  velocity,  w,  the  relation  between  the  linear 
velocity  of  the  particle  and  the  angular  \elocity  may  thus 
be  found. 


(J) 

uniform   or 

)y  the  ladiiis 

is  describetl ; 

elocity,  take 

[I  tj'ied  point, 

sfl  of  angular 
ayured  in  tlie 
'liii-s  wlietlier 
al)le,  it  may 
finite.simal  of 
locity  will  be 
at  any  time, 


(2) 

r  velocity  is 
s  the  increase 

the  preceding 


I  line  at  the 
olves  with  a 
?en  the  linear 
ity  may  thns 


EXAMPLES. 


277 


Let  d6  be  the  angle  throiigli  wliich  the  radius  revolves  in 
the  time  dt,  ai'd  let  dx  l)e  the  path  described  by  the  particle, 
so  that  ds  =  rdO  ; 


then 


ds  do 


■;  )  that  the  linear  velocity  varies  as  the  angular  velocity  and 
•  he  length  of  the  radius  jointly. 

2.   If  the  angular  acceleration  is  a  cons^lnt,  as  (p ;  then 
from  {'^)  we  have 


</26> 


df' 


1  =  0; 


dd 
dt 


^   V  +  '-'o. 


and 


e  =  ^^/'  +  i.,J  +  0„, 


where  w^  and  6^  are  the  initial  values  of  u)  and  9. 

Hence  if  a  line  revolves  from  rest  with  a  constant  angular 
acceleration,  we  have 

6  =  k/</2 ; 

and  the  angle  described  by  it  varies  as  the  s(|uari.  of  the 
time. 

'].  If  a  particle  revolves  in  a  circle  uniformly,  and  its 
place  is  continually  projected  on  a  given  diameter,  tiie 
linear  acceleration  along  that  diameter  varies  directly  as 
the  distance  of  the  projected  place  from  the  centre. 

Ijet  0)  be  the  constant  angular  velocity,  9  the  angle 
between  the  fixed  diameter  and  the  radius  drawn  from  the 
c'.'Mtre  to  its  place  at  the  time  /,  x  the  distance  of  this 
projected  place  from  the  centre.  Then,  calling  a  the 
radius  of  the  circle,  we  have 

iC  =  (I  cos  0, 


278 


EXAMl'LKS. 


dx  .     ^dO  .     „ 

-—  =  —  «  Sin  0  ,.  =  —  flw  sin  d; 
at  dt  ' 


d^ 


do 


whicli  proves  the  theorem. 

4.  If  tlie  angular  acceleration  varies  as  tiie  angle 
generated  from  a  given  fixed  lint,  and  is  negative,  find  tlie 
angle. 

Here  the  equation  wliich  exju-esses  the  motion  is  of  tlie 
form 

§  =  -»■ 


Calling  «  the  initial  value  of  d  we  find  for  the  result 
0  =  ic  cos  kt. 

5.  If  a  particle  revolves  in  a  circle  with"  a  uniform 
velocity,  sliow  that  its  angular  velocity  about  any  point  in 
tlie  circumference  is  aNo  uniform,  and  ecjual  to  one-half  of 
what  it  is  about  the  centre. 

At  jtrescnt  this  is  sufticient  for  the  general  explanation 
of  angular  velocity  and  angular  acceleration.  We  shall 
return  to  tlie  subject  in  Chap.  7,  Part  III.,  when  we  treat 
of  the  motion  of  rigid  bodies. 

161.  The  Component  Accelerations,  at  any  instant, 
Along,  and  Perpendicular  to  the  Radius  Vector.— 

Let  (/',  0)  (Fig.  79)  be  the  place  of  the  moving  iiartidc,  I\ 
at  the  time  /,  {.t;  //)  being  its  place  referred  to  a  system  of 
rectangular  axes  having  the  same  origin,  and  (he  r-axis 
coincident  with  the  initial  line.     Then 


x  =  r  cos  0\   1/  ~  r  sin  S ; 


(1) 


tlie    iuigli." 
ive,  find  tlio 

:»ii  is  of  tlic 


e  result 


:i  unifoi'in 
any  point  in 
)  ono-hiilf  of 

cx])laniili()n 

We   sliiill 

len  we  treiit 


ny  instant, 
I  Vector. — 

|)iirti(l(\  P, 
a  system  of 
i  (lie  .r-iixis 


(1) 


RADlCAh  AND   TRANSVKRHAL  ACCELERATION'S.    279 


tiierefore 

and 

fPx rcPr 

Similarly 


dx       dr        ^  ■    r^dd 

^  =  ^co8e~rBm0^; 


(3) 


©■]-»-[> 


.dr    dd   ,      (PS-\   .    a  ,n^ 
'di-di+'dPr''^-^^^ 


(Pdl 


which  are  the  accelerations  parallel  to  the  axes  of  x  and  y. 
Resolving  these  along  the  radius  vector  by  multiplying  (3) 
and  (4)  by  cos  B  and  sin  0  resjjectively,  since  accelerations 
may  be  resolved  and  conij)ounded  along  any  line  the  same 
as  velocities  (Art.  149),  and  adding,  we  have 


dt^ 


^  ,  (Py   .    „       (Pr         /ddY  ,.. 

cos0  +  ^sm0  =  ^p-ry;  (5) 


which  is  the  acceleration  along  the  radim  vector.* 

Multiplying  (3)  and  (4)  by  sin  6  and  cos  0  respectively, 
and  subtracting  the  former  from  the  latter,  we  get 


(Py 
dP 


cos  d 


(Px   .    „        ,  dr    de         d?d 

dP  ^'"  ^  =  ^  dt  •  It  -'  '■  rfr» 


_  1  rf  /  2  </6i\ 
~  r  dt  V'  dtr 


(6) 


which  IS  the  acceleration  perpendicular  to  the  radius  i'rctor.\ 

162.  The   Component   Accelerations,  at  any  in- 
stant, Along,  and  Perpendicular  to  the  Tangent— 

U't  (j;  y)  (Fig.  79)  be  the  place  of  the  moving  particle.  I', 
at  tiic  time  /,  and  .v  Ihc  Icnulli  of  the  arc  described  during 


•  Hiinictliiu'H  lalli'd  Ih,.  lailiui aceekra'ion. 
t  Sonietlmeii  callud  the  traniiv«r»(U  aectltfalUm. 


ita 


280 


TANGENTIAL   ACCELERATION. 


that  time.    Then  the  iici'ek'rations  along  the  axes  of  x  and  «/ 

JO  JO  7  1 

iii'e  -7^  and   yf ;  and  the  direction  cosines*  are    ,-  and  ~^- 
ur  at'  lis  Us 

To  find  the  acceleration  along  the  tangent  we  must  multi- 
ply these  axial  accelerations  by    .^  and  '-f,  respectively,  and 

(IS  CIS 

add.     Thus  the  tangential  acceleration,  T,  is 


„  _  ^    dx       iPy    iy 
dfl  '  ds  ■•"  rf/2  ■  ds ' 


(1) 


Since  d»^  =  dT>  +  dy\  therefore,  by  differentiation  we 
have 

ds  (Ps  =  dx  {fix  +  dy  d?y ; 

and  dividing  by  ds  dP  we  get 

iPs  _  (Px    dx       fPy    dy 


dfi 
which  in  (1)  gives 


dP  '  ds  +  dP  '  ds' 


~  dl^' 


(2) 


for  the  acceleration  alony  the  tangent. 
Similarly  we  have  for  the  normal  acceleration,  N, 

~Wi'ds~  dt^  '  ds 

__  (d^y  dx  —  (Px  dy)    d^ 
~  'd?  dp 

1    rfs* 
=  -  •  -A3 ,  (by  Ex.  4,  p.  144,  Calculus), 

where  p  is  the  radiu.s  of  curvalure  ; 

*  C'Dvliiuii  of  Ihi'  aiiijlus  which  thu  tiiii^'oiit  inukos  with  tlie  axes  of  r  and  y. 


;es  of  X  and  «/ 

dx      ,  du 
e    ,-  una  -p- 
</s  dt( 

must  niiilti- 
)ectively,  and 


(1) 


entiation  we 


(2) 


n,N, 


,  Calculus), 


!«  of  r  and  y. 


NORMAL   A  C(  KI.EKA  TWA. 


P 


281 
(3) 


vf :;  is  ihr  vplocity  of  the  particle  at  the  point  (x,  y). 

Hence  ut  any  jwint,  P,  of  the  trajectory,  if  the  accelera- 
tion is  resolved  along  the  tangent  to  the  curve  at  P  and 
along  the  normal,  the  accelerations  along  the  two  lines  are 
respectively 

df^  '^"^  r 

163.  When  the  Acceleration  Perpendicular  to 
the  Radius  Vector  is  zero.— Then  from  (U)  of  Art.  161 
we  have 


,dd 
'  dt 


constant  =  h  suppose; 


and 


dd 
•*•  di- 

h 

dr 
dt 

dr    do 

~  dd  '  dt 

h 

-  ri 

dr 

'  de' 

dfi  ~  r*'  d0i      ^  1^  \dBI ' 


which  in  (5)  of  Art.  161  gives 

the  acceleration  along  the  radius  vetttor 


,.4  \((p 


/•MrfO/       r8' 
an  expression  which  is  indei^ndent  of  /. 


(1) 


'his  may  he  put  into  a  more  convenient  form  as  follows: 


\H  r 

=     ;  then 
u 

dr            1     du _ 

de        M» '  rfe ' 

26i 


CONSTANT  A.\(1VLAI{     VKLOVITY. 

'   dm  ~      ?<2  ■  d(P  '^  1?  \dd} ' 


wliich  in  (1)  and  reducing,  gives 

tiie  acceleration  along  the  radius  v<»oior 


(-•) 


P^rom  these  two  formulae  the  law  of  acceleration  along 
the  radius  vector  may  be  deduced  when  the  curve  is  given, 
and  the  curve  may  be  deduced  when  the  law  of  accelera- 
tion along  the  radius  vector  is  given.  Examples  of  these 
processes  will  be  given  in  Chap.  {-Z),  Part  III. 

164.  When  the  Angular  Velocity  is   Constant— 

Let  the  angular  velocity  be  constant  =  w  suppose.     Then 

dd 
di=''* 

therefore  from  (5)  of  Art.  161 

the  acceleration  along  the  radius  vector 

The  acceleration  perpendicular  to  the  radius  vector 

dr 


(1) 


=  2(i) 


df 


(^) 


and  both  of  these  arc  indei)endent  of  9. 

The  following  example  is  an  illustration  of  these 
formula' : 

A  particle  describes  a  i)ath  witii  a  constant  angiihir 
velocity,  and  without  acceleration  ahnig  the  radius  vector; 
find  (1)  the  equation  of  the  path,  and  (2)  the  acceleration 
perpendicular  to  the  radius  vector. 


r:-^!*^mmai 


ration  along 
ve  is  given, 
of  accelera- 
k's  of  tliese 


'onstant— 

)se.     Then 


^^ector 


(1) 


(2) 


n    of    tliosc 

lilt  iiiigiilar 
lius  vt'ctor; 
acceleration 


C0ySTA.\7'  ASdULAR    VELOCITY. 


288 


(1)  From   (1)   w"  have,   from    the    conditions    of    the 
question. 

Integrating  we  have 


w2r  —  0. 


dfi 


—  0)8  (r*  _  a?)f 


dr 


\t  r  =  a  when  -3-  =  0. 
at 


Therefore 


dr 


=  o)dt', 


(r2  _  r,2)i 
if  r  =  o  when  t  =  0, 


.'.    r  =  5(r'  +  e-"'). 


(3) 


de 


Also,  as  Yf  =  o),  therefore  6  =  oit,  it  6  =  0  when  <  =  0. 


Substituting  this  value  of  u)t,  we  have, 


r  =  ,^{e»  +  e-o) ; 
which  is  the  path  described  by  the  jiarticlc. 


(*) 


{'i)  Tjet  Q  be  the  required  acceleration  perpendicular  to 
the  radius  vector,  then  from  (2)  we  have 


Q^^ii 


dr 
(it 


uu)^  U^  —  t'-"').  fi'om  (3) 


'4S4: 


EXAMPLES. 


=  aw2  (e»  —  e-9) 


(5) 


wliich    is    the  acceleration    perpendicular    to    tlie   radius 
vector. 

The  preceding  discussion  of  Kinematics  is  snfticient  for 
this  work.  There  are  various  other  problems  whicli  might 
be  studied  as  Kinematic  questions,  and  inserted  here  ;  but 
we  prefer  to  treat  them  fioni  a  Kinetic  point  of  view. 

For  the  investigation  of  the  kinematics  of  a  i)articie 
describing  a  curvilinear  path  in  space,  see  Price's  Anal. 
Mech's,  Vol.  I,  ]).  430,  also  Tait  and  Steele's  Dynamics  of 
a  Particle,  p.  12. 


EXAMPLES. 

1.  A  particle  describes  the  hy])erbola,  xy  =  P;  find  (i) 
the-  acceleration  i)arallel  to  tiie  axis  of  x  if  the  velocity 
parallel  to  the  axis  of  //  is  a  constant,  (3,  and  (2)  find  the 
acceleration  iiarallel  to  the  axis  of  //  if  the  velocity  parallel 
to  the  axis  of  a:  is  a  constant,  «. 

?^.^-  (2)  ^"' 


A  «.v, 


(i)=^^-«;  (a)  -i^f 


2.  A  ])artic]e    lescribes   the   paral)ola,    ?/»  =  4a.r  •    find 

tile  acceleration  parallel  to  the  axis  of  //if   the   velocity 

l)arallel  to  the  axis  of  x  is  a  constant,  «.  4^,v 

A  U.S.   — 

:5.  A  particle  describes  the  logarithmic  curve,  //  z=  a-^; 
find  (1)  the  .f-component  of  the  acceleration  if  the  y-com- 
ponent  of  the  velocity  is  a  constant,  fi,  and  (2)  find  the 
//-component  of  the  acceleration  if  the  r-comi)onent  ..f  the 
velocity  is  a  constant.  «. 


^"'-   ^'^-«^log«'  (•^)«''0»g«)^y- 


EXAMPLES. 


285 


(5) 
the   riKlius 

iiftioioiit  for 
liicli  might 
(I  here  ;  but 
P  view. 

)f  a  particio 
Vice',s  Anal. 
Dynamics  of 


^•2;  find  (1) 
the  velocity 
(S)  find  the 
city  parallel 

'ifi-r  ;    find 
lie    velocity 

t 

e,  y  —  a' ; 
the  y-com- 
2)  find  the 
lenl    of  (he 


\ogay-y. 


4.  A  particle  describes  the  cycloid,  the  starting  point 
being  the  origin;  liud  (1)  the  j-cornponent  of  the  accel- 
eration if  the  //-component  of  the  velocity  is  /3,  and  (2)  find 
the  _y-component  of  the  acceleration  if  the  x'-comptnient  of 


the  velocity  is  «. 


Ahs.  (1/ 


li^ay 


{■Zay 


„     8    ' 


(^)- 


5.  A  particle  describes  a  catenary,  y  =     ic"  -}-  <•    "1; 

find  (1)  the  . '-component  of  tlie  acceleration  if  the //-com- 
ponent of  the  velocity  is  (i,  and  (2)  find  the  ^-component 
of  the  acceleration  if  the  r-compoiient  of  the  velocity  is  «. 


yins. 


(I)__^«^_.. 
(//-«')«' 


i^)ty. 


6.  Determine  how  long  a  particle  takes  in  moving  from 

the  point  of  projection  to  the   furtlur  end  of   the  latns 

rectum.  .        /•  ,  .  , 

Atifi.  -  (sin  «  +  cos  «). 

// 

7.  A  gun  was  fired  at  an  elevation  of  50°;  the  ball 
struck  the  ground  at  the  distance  of  244!)  ft.;  find  (1)  the 
velocity  with  which  it  left  the  gun  and  (2)  the  time  of 
flight,    {g  =  32i). 

Alls.  (I)  282.8  ft.  per  sec;  (2)  13.47  sees. 

8.  A  ball  fired  with  velocity  n  at  an  indiiuition  «  to  the 
horizon,  just  clears  a  vertical  wall  which  subtends  an  angle, 
(i.  at  the  point  of  projection;  determine  the  instant  at 
which  the  ball  just  clears  the  wall. 

u  sin  «  —  I/// 

U  cos  rt 


Ans. 


tan  (i. 


5).   In  the  preceding  example  determine   tiie  horizontal 
distance  between  the  foot  of  the  wall  and   the  i)oitit   where 


the  bill!  strikes  the  ground. 


Aths.     —  cos*  «  tan  H. 
U 


286 


EXAMPLES. 


10.  At  the  (listunee  of  a  quarter  of  ii  mile  from  the  bot- 
tom of  ii  dirt",  wliicli  is  120  It.  liigli,  a  shot  is  to  be  lireil 
which  sliall  just  clear  the  cliff,  and  pass  over  it  horizon- 
tally ;  find  the  angle,  «,  and  velocity  of  projection,  v. 

Ans.   a  =  10"  18';  ?•  =  490  ft.  per  sec. 


the   range    i- 


11.    When    the   angle  of  elevation    is   40 
2449  ft. ;  find  the  range  when  the  elevation  is  29f' . 

Ans.  2131.5  ft. 


12.  A  body  is  projected  horizontally  with  a  velocity  of 
4  ft.  ])er  sec:  tind  the  latiis  rectum  of  the  parabola  de- 
scribed, (//  =  ;}•>).  Ans.   1  foot. 

13.  A  body  projected  from  the  top  of  a  tower-at  an  angle 
of  45"  above  the  horizontal  direction,  fell  in  5  sees,  at  a 
distance  from  the  bottom  of  the  tower  equal  to  its  altitude; 
find  the  altitude  in  feet,  {(/  =  32).  Ans.   200  feet. 

14.  A  ball  is  fired  up  a  hill  whose  inclination  is  15°; 
the  inclination  of  the  i)iece  is  4.V',  and  the  velocity  of  pro- 
jection is  500  ft.  per  sec;  find  the  time  of  flight  before 
it  strikes  the  hill,  and  the  distance  of  the  place  where  it 
falls  from  the  point  of  projection.* 

Alls.  T  =  10. n  sees.;  R  =  1.121  miles. 

15.  On  a  descending  jjlane  whose  inclination  is  12°,  a 
l)all  fired  from  the  top  hits  the  plane  at  a  distance  of  two 
miles  and  a  half,  the  elevation  of  the  piece  is  42°  ;  find  the 
velocity  of  projection.  Ans.  v  =  579.74  ft.  per  sec. 

K).  A  body  is  projected  at  an  inclination  «  to  the  hori- 
zon :  determine  when  the  motion  is  i)erpendicular  to  a 
plane  which  is  inclined  at  an  angle  ti  to  the  horizon. 

,        n  sin  <(  —  (if. 
A  US. •'    —  ±  cot  3. 

II   cos  (C  -L.  r- 


'  The  ruiigf  on  tlie  iiirlincil  plane. 


II 


EXAMPLKS. 


2S7 


rom  tl'e  hot- 
is  to  be  liroil 
f  st  horizon- 
tion,  V. 
't.  per  sec. 

lie   raiigf    i.-, 

'.or- 

2131.5  ft. 

1  velocity  of 
parabola  de- 
is.   J  foot. 

•at  an  an<(le 
5  seta,  at  a 
its  altitude; 
200  feet. 

tion  is  15°; 
city  of  pro- 
light  heforo 
ce  where  it 

.21  miles. 

)n  is  12°,  a 
nice  of  two 
° ;  find  the 
'..  per  sec. 

0  the  liori- 
icular  to  a 
izon. 

±  cot  (i. 


17.  Calculate  the  maxiuuiin  range,  and  time  of  flight, 
on  a  descending  iilane,  the  angle  of  depression  of  which  is 
15°,  tlie  velocity  of  projection  being  1000  ft.  per  sec. 

Ann.   Max.  range  =  7.it8  miles  ;  T  =  51.34  sec. 

18.  With  what  velocity  does  the  ball  strike  the  plane  in 
the  last  example  ?  Ans.    V  =  VMY.i  feet. 

lit.  If  a  ship  is  moving  horizontally  with  a  velocity 
=  ;](/,  and  a  body  is  let  fall  from  the  top  of  the  mast,  find 
its  velocity,  V,  and  direction,  0,  after  4  sees. 

Auk   V  =  5//;  0  =  tan  '  ^. 

20.  A  body  is  jirojected  iiorizontally  from  the  top  of  a 
tower,  with  the  velocity  gained  in  falling  down  a  space 
eipial  to  the  height  of  the  tower;  at  what  distance  fron\ 
the  base  of  the  tower  will  it  strike  the  gronml  ? 

Ans.   R  =  twice  the  height  of  the  tower. 

21.  Find  the  velocity  and  time  of  flight  of  a  body  pro- 
,jecled  from  one  extremity  of  the  base  of  an  C(iuilateral 
triangle,  and  in  the  direction  of  the  side  adjacent  to  that 
extremity,  to  pass  through  the  other  extremity  of  the  base. 

,  .  /-«.'/.  'p  _  .  /2rtV3 

V  v';5  V      c/ 

22.  Given  the  velocity  of  sound,  V;  find  the  horizontal 
range,  when  a  ball,  at  a  given  angle  of  elevation,  «,  is  so 
jirojectcd  towards  a  jicrson  that  the  ball  and  sound  of  the 
discharge  reach  him  at  the  same  instant. 

A  US.   "       tan  «c. 
// 

23.  A  body  is  projected  horizontally  with  a  velocity  of 
4f/  from  a  point  whose  height  al)ove  the  ground  is  Ki/y  ;  tiiid 
tiie  direction  of  motion.  ^>.  (!)  when  it  has  faUeii  iiali'-way 
to  the  ground,  and  (2)  when  half  the  whole  time  of  falling 
hag  elapyed.  -     -1 


.l,,.v.   (1)«  =  45°;  (2)0 


tan 


Vi 


288 


EXAMl'LES. 


\ 


^4.  Particles  are  projectod  witli  a  given  velocity,  /•,  in 
all  lines  iii  a  vertical  i)lane  from  tiie  point  0  ;  find  the  locus 
of  them  at  a  given  time.  /. 

Aus.  x^  +  (ii  +  V/y/-')^  =  vVK  whicii  is  the  equation  of  a 
circle  whose  radius  is  rt  and  whose  centre  is  on  the  axis  of 
//  at  a  distance  hjl'^  below  the  origin. 

2.").  How  much  powder  will  tiirow  a  i;}-inch  shell* 
•tOOO  ft.  on  an  inclined  plane  wIdsc  angle  of  elevation  is 
10°  -10';  the  elevation  of  the  mortr.r  being  '.]b\ 

Alts.  Charge  =  4.07  lbs. 

20.  A  ju-ojectilc  is  discharged  in  a  horizontal  direction, 
with  a  velocity  of  450  ft.  j)er  sec,  fn»m  tlie  summit  of  u 
conical  hill,  the  vertical  angle  of  which  is  120  ;  at  what 
distance  down  the  hillside  will  the  projectile  fall,  and  what 
will  be  the  time  of  flight? 

Alls.   Distance  =  2812.0  yards ;  Time  =:  10.23  sees. 

2?.  A  gun  is  placed  at  a  distance  of  500  ft.  from  the  base 
ofa  cliff  which  is  200  ft.  Iiigh  :  on  the  edge  of  the  cliff 
there  is  built  the  wall  of  a  castle  OO  ft.  high  ;  find  the 
elevation,  «,  of  the  gun.  and  the  velocity  of  discharge,  r, 
in  order  that  the  ball  may  graze  the  top  of  the  castle  wall, 
and  fall  120  ft.  inside  of  it. 

Alls,   a  =  5.3°  VS  ;  /•  =  105  ft.  per  sec. 

28.  A  piece  of  ordnance  bui-st  when  50  yards  from  a 
wall  14  ft.  high,  and  a  fragment  of  it.  oi'iginally  in  con- 
tact with  the  ground,  after  grazing  the  wall,  fell  0  ft. 
beyond  it  on  the  oi)j)osite  side  ;  find  how  high  it  rose  in 
fli^"  'lir-  AnR.  U  ft. 


*  The  weight  of  a  13-lncli  shell  Is  196  lbs. 


velocity,  i\  in 
lind  tile  locus 


^ 


t'<jiiation  of  a 
m  tlie  axis  of 


5-incli    shell* 
f  elevation  is 


PART    III. 

KINF.TICS   (MOTION   AND   FORCE), 


=  4.07  lbs. 

tal  direction, 
summit  of  u 
iO  ;  at  what 
ill,  and  what 

10.23  sees. 

Vom  the  base 
of  the  cliff 
<i\\ ;  find  the 
discharge,  v, 
?  castle  wall, 

ft.  ])er  sec. 

ards  from  a 
iially  ill  con- 
11.  fell  0  ft. 
gh  it  rose  in 
iriR.  94  ft. 


CHAPTER     I . 

LAWS  OF  MOTION— MOTION  UNDER  THE  ACTION  OF 
A  VARIABLE  FORCE- MOTION  IN  A  RESISTING 
MEDIUM. 

165.  Definitions.— /I'l^eZ/Vv  is  thttt  branch  of  Dynamioi, 
which  treats  of  the  motion  of  bodies  under  the  action  of 
forces.  ■ 

la  Part  I,  forces  were  considered  with  reference  to  the 
pressures  which  they  jirodut-ed  upon  bodies  at  rest  (Art. 
15),  i.  e.,  bodies  under  the  action  of  two  or  more  forces 
in  equih  rium  (Art.  20).  In  Part  II  we  considered  the 
purely  geometric  properties  of  the  motion  of  v.  point  or 
j)article  without  any  reference  t(»  the  causes  producing  it, 
or  the  iiroperties  *)f  the  thing  moved.  We  arc  now  to 
consider  motion  witli  reference  to  the  causes  which  produce 
it,  and  the  things  in  which  it  is  produced. 

Tlie  student  must  here  review  Chapter  T,  Part  I,  and  obtain  rlear 
conceptions  of  Mumentum,  Accileratinn  of  Momentum,  and  the  Kiiietir 
meosnre  of  Foref  (Arts.  12, 13, 19,  sind  20),  as  this  is  necessary  to  a  full 
iiiiderstandinff  of  tl)e  fundnniental  laws  of  motion,  on  the  trtitti  of 
which  all  our  succeedintf  invest ifrations  are  founded. 

166.  Newton's  Laws  of  Motion.— The  fundamental 
13 


Em 


2'JO 


XEtyroys  imwn  of  Mono. \. 


})riiicii)lo;i  in  lueKnlaiicL-  with  wliicli  inotiuii  takes  place  are 
fiiiboilicd  ill  throe  stutonienta,  generally  known  an  Mirfon's 
Jmh'i  of  Motion.  These  laws  nnisl  he  eonsidered  as  resting 
on  con viel ions  drawn  from  ohservation  and  experiment, 
and  not  on  intuitive  perception.*     The  laws  are  the  fol- 


Law  L— Every  hndij  contimtes  in  its  state  of  rest 
or  of  itiiiform  niotiau  in  a  straight  line,  except  in 
so  far  as  it  is  conipclied  by  force  to  change  that 
state. 

Law  W.— Change  of  motion  is  proportional  to  the 
force  applied,  ami  takes  place  in  tlie  direction  of 
the  straight  line  in  adiich  the  force  acts. 

Law  Ul.—To  cilery  action  there  is  always  an 
eqaal  and  contrary  reaction:  or.  the  niiitaal  ac- 
tions of  any  two  bodies  are  always  eqaal  and  oppo- 
sitely directed. 

167.  Remarks  on  Law  I.— Law  I  suppii.M  u«  with  a 

(lofiiiition  (if  loic<'.  It  iiulicatcs  tliat  force  i.s  that  whicii  tends  to 
change  a  bodj'H  state  of  rest  or  of  miiforni  motion  iu  a  straiglit  line  : 
for  if  a  hody  docs  not  continue  in  its  state  of  rest  or  of  uniform  mo- 
tion in  a  straight  line  it  must  be  under  the  action  of  force. 

A  body  has  no  power  to  rhange  its  own  state  as  to  rest  or  motion  ; 
when  it  is  at  rest,  it  lias  no  ])ower  of  jmttiiig  itself  in  motion  ;  wlieu 
in  motion  it  has  no  power  of  increasing  or  diminishing  its  velocity. 
Matter  is  inert  (Art.  3).  If  it  is  at  rest,  it  will  remain  at  rest ;  if  it  is 
moving  with  a  given  velocity  along  a  rectilinear  path,  it  will  continue 
to  move  with  that  velocity  along  tlmt  path.  Ft  is  alike  natural  to 
matter  to  l)e  iit  rest  or  in  motion.  Whenever,' therefore,  a  body's 
state  is  changed  either  from  re,-t  to  motion,  or  from  motion  to  rest, 
or  when  i;.<  velocity  is  increase  I  or  diminished,  that  change  is  due  1 1 
some  external  cause.  This  cause  is  I'lilliul  fmt'  (Art.  14);  and  the 
W(>i<l/;>/w  is  used  in  Kinetics  in  tliis  nuaniug  only. 


•  TUonii-o    ajid  TaitV  Nat.  Pliil.,  p.  ^1. 


kes  place  are 

1  as  Newtoii'H 

ed  as  resting 

experinu'iit, 

are  the  I'ol- 


tdte  of  rest 
,  rxci'pt  in, 
'la/iQ'e  that 


oikU  to  the 
Lrection  of 
f. 

ilwaijs  ail 
iiUioal  ar- 
'  and  oppo- 


t'M  US  with  a 
liic'li  tends  to 
.straijtflit  line  ; 
'  uniform  nio- 
•ce. 

'St  or  motion  ; 
notion  ;  wlicn 
I  its  velocitj-. 
I  rest  ;  if  it  is 
.  will  continue 
ike  natural  to 
fore,  u  body's 
lotion  to  rest, 
nfje  is  due  1 1 
14) ;   and  tlie 


i 
I 

^ 


/./•..l/.l/.'A.S    O.V    LAW  II. 


X»!tl 


168.    Remarks  on  Law  II. — Law    II   asserts  that  if  any 
,niri.  jrenerates  motion,  a  double  force  will  generate  doulile  motion, 
niid  so  on,  wh.'thor  applied  simultaneously  or  successively,  instau 
i;ineously  or  gradually.     And  this  motion,  if  the  Ixxly  was  movintf 
I)  forehand,  is  either  added  to  the  previc-us  motion  if  directly  cons]  ir 
iujf  with  it,  or  is  subtracted  if  directly  oi)poscd  ;  or  is  geometrically 
conipouiided   with  it  according  to  the   principles  already  explaiiuvl 
(.\rt.  29l,  if  tlie  lino  of  previous  motion  and  the  directi(jn  of  the  f..rce 
are  im-liiied  to  eucli  other  at  an  angle.     The  term  7Witi»n  here  meai.s 
qiiaidity  of  motion,  and  the  phrase  c/(^(//i/e  o/ «"<'"«  here  means  rnte 
iijChnnyc  of  qniintity  of  nwlioii  lArt.  1:5).     If  the  force  be  Unite  it  will 
require  a  finite  time  to  produce  a  sensible  change  of  motion,  and  the 
change  of  momentum  pHxluccd  by  it  will  depeuil   upon  the  time  dur- 
ing  -.vhich  it  acts.     Tlu?  change  of  motion  must  then  be  understood  tir 
be  the  change  of  momentum  produced  per  un  t  of  time,  or  the  rut 
of  change  of  momentum,  or  nccelerati..ii  of  momentum,  whicli  agrees 
with  the  principles  already  explained  (.Arts.  13  and  20).     In  tiiis  law 
nothing  is  said  about   the   actual   motion  of  the  body  before   it  was 
acted  on  by  the  force;  it   is  only  the  c/cn/.'/c  of  iiiolion  that  concerns 
us.     The  same  force   will   i)roduce  precisely  the  same  change  of  mo- 
tion in  a  body;  whether  the  bwly  be  at  rest,  or  in  motion  with  any 
velocity  whatev(-r. 

Siiifo,  wlu'ii  .*('viM-al  furees  act  at  oiicl'  oii  u  particle  either 
at  rest  or  in  motion,  the  .sceoii'l  law  of  motion  is  true  for 
erery  ouv  of  these  forces,  it  follows  tliat  each  must  htive  the 
same  etfect.  in  so  far  as  the  chaiiac  of  motion  i)riHlucecl  l)y 
it  is  concerneil,  as  if  ;/  were  the  only  force  in  action. 
Hence  the  assertion  of  the  secoiul  law  may  be  put  in  tiie 
following  form  : 

Wlun  (1)11/  nuinho-  of fo ires  ort  simuHaneotn^hj  on  a  hody, 
ivhellier  at  rrnf  or  in  motion  in  any  dirrrtion.  mrli  forrr  pro- 
i/iir<'s  in  the  hoihj  llie  sanu-  c/mnyo  of  motion  as  if  it  alone 
had  acted  on  llic  hody  at  rest. 

It  follows  from  tliis  view  of  tlie  law  that  all  problems 
which  involve  forces  actiiifr  siiiiiiltancoiisly  may  be  inaicil 
as  if  the  forces  acted  .snrcrssirrJy. 

The  operations  of  this  law  have  p' ready  been  considered  in  Kine 


292 


REM  AUKS   OX  LaW  IT. 


Jiiatics  (Art.  149) ;  but  motion  there  was  understood  to  mean  velocity 
only,  since  tlie  maes  of  tlie  IkkIv  was  not  considered.  Tbis  law  in 
eludes,  therefore,  the  law  of  the  coniijosition  of  velocities  already 
referred  to  (Art.  29).  Another  consequence  of  the  law  is  the  follow- 
ing :  Since  forces  an;  measured  by  the  changes  of  motion  they  |  ro- 
(luce,  and  their  directions  assigned  by  the  directions  in  which  the.se 
dianges  are  produced,  and  since  the  changes  of  motion  of  one  and  tlie 
i;aine  body  are  in  the  <lirectiona  of,  and  proportioniil  to,  th((  changes 
of  velocity,  therefore  m  single  force,  measured  hy  the  resultant  change 
of  veloci*y,  and  in  its  direction,  will  be  the  equivalent  of  any  number 
of  simultaneously  acting  forces. 

IloilfO, 

The  rcsidiaitf  of  an//  number  of  concurrin(/  forces  is  to  be 
found  by  the  same  (jeomeirie  process  as  the  resultant  of  any 
number  (f  simuJlaneovs  velocities,  and  conversely. 

From  this  follows  at  once  tlio  J'oly/jon  of  Velocities  aiifl 
the  Parallelopiued  of  Velocities  fi-oni  the  Polygon  and 
Parallcloijipod  of  Forces,  as  was  described  in  Art.  14:;'. 

This  law  also  gives  us  the  means  of  measuring /--m-,  and  ahso  of 
measuring  the  maitii  of  a  lio<ly  :  for  the  actions  of  diflereiit  forces  upon 
the  same  body  for  e(|ual  times,  evidently  produce  <'!ianges  of  velocity 
wliich  are  proporliioia'  to  tin  ./'.(/rev  Also,  if  eipud  forces  act  on  dif 
f'rent  IxMlies  for  e(|ual  times,  the  changes  of  velocity  i)ro(luced  must 
be  iiirersi''if  as  the  mr/.«.v,,v  of  the  bodies.  Again,  if  difterent  bodies, 
each  acted  on  by  a  force,  accpiire  in  tli<^  same  time  the  same  changes 
of  velocity,  the  forces  must  bi'  ])roi)ortlonal  to  the  masses  of  the 
bodies.  This  means  of  measuring  force  is  i)ractical]y  the  same  as 
that  already  deduced  by  abstract  reasoning  (Arts.  1!)  and  30). 

It  appears  from  this  law.  thjit  every  theorein  of  F^iiie- 
matics  coiiiu'cted  with  acceleration  litis  its  counterpart  in 
Kinetics.  'riuis.  the  mcnsiire  (if  acceler;if ion  or  velocity 
increment,  (.\rt.  '.)).  wiiich  was  discn.ssed  in  Chap.  I  (Arl,<. 
H   and    H).   and    in    Kinemiitics   (.\rt.  i:;.')).    and    which  is 

denoted    hy   /'or   ils   e(|iial     ,'„.  is   al.so  liie  ell'ect    and   the 

III- 

mciisiiro  of  force  ;  tiu'refore  all  liie  icsnlls  of  the  e<iiia(ion 


KKMAIiKS    o.V    LA  II'    //. 


B  mean  velocity 
.  Tills  law  lii- 
loclties  already 
\'  is  the  follow- 
ofion  they  |  ro- 
iii  wliicli  tlicso 
of  one  1111(1  tli(> 
I),  tli((  cliungcs 
isullaiit  change 
of  any  number 


'orces  is  to  be 
(It (1)1 1  of  any 

Velocities  and 
'(•lygoii   and 

er,  and  also  of 
lit  forces  ii])on 
j^es  of  velocity 
oes  act  on  dif. 
)roduce(l  must 
ft'erent  bodies, 
sanie  changes 
masses  of  the 
y  the  same  as 
120). 

!n  of  r^iiic- 
intcrparl  in 
DP  vi'lot'ity 
Imp.  r  (Arts, 
id    wliioli  is 

'ft   and   till' 

I'  i'(ination 


(1) 


its  various  forms,  and  the  reniari<s  wliich  have  been  made 
on  it,  are  applicalile  to  it  when  ./'  is  tiie  accelerating  force. 
Thus,  (Art.  10'^),  we  sec  tliat  the  force,  under  wiiich  a 
particle  describes  any  cnrve,  may  l)e  resolved  into  two 
components,  one  in  the  tangent  to  the  curve,  the  other 
/o/rfl/(/,s  the  centre  of  curvature:  liieir  nuignitudes  being 
the  acceleration  of  momentum,  and  the  product  of  the 
momentum  into  the  angular  velocity  about  the  centre  of 
curvature,  respectively.  In  the  case  of  uniform  motion, 
the  first  of  these  vanishes,  or  tiie  whole  force  is  perpen- 
dicular to  the  direction  of  motion.  VViien  there  is  no  force 
perpendicular  to  tiie  direction  of  motion,  there  is  no  curva- 
ture, or  the  path  is  a  straiglit  line. 

Hence  if  we  suppose  the  particle  of  mass  w  to  be  at  the 
point  (.(•.  ;(/.  z),  and  resolve  the  forces  acting  on  it  into  the 
three  rectangular  eomj)onents,  X,  Y,  Z,  we  have 


in 


dPx 
tlt^ 


X 


in 


iPz 
(IP 


(3) 


In  several  of  the  chapters  these  ecpititions  will  be  sim- 
l)litied  bv  assuming  unity  as  the  mass  of  the  moving 
particle.  When  tliis  cannot  be  done,  it  is  sometimes  cim- 
venient  to  assume  X.  Y,  Z,  as  the  component  .orces  on  the 
unit  muss,  and  {-l)  l)ecomes 

m  -r-,  —  niX,  etc. 
dr 

from  which  m   may  of  course  be  omitted.     It  will  be  ob- 
served that  an  equation  such  as 

(Px 


(U^ 


=  X 


may  be  interpreted  either  as  Kinctical  or  Kinematical;  if 


a'.)4 


lihWAUKS   OS   I, A  W  III. 


tlie  former,  tlic  unit  of  muss  must  bo  undcrstoou  as  a  liio- 
•or  on  till'  li'l'tliaiid  siilo,  in  wiiicii  case  X  is  liic  ./-ooni- 
ponent.  i'<ir  I'li'  unit  of  mass,  of  [\\v  wliolc  t'oiro  '  xorled  (<ii 
till'  niuvi'^  l)oily. 

'I'!'?  lirst  two  liiwH,  Imvo,  tlieroforc,  liirnislicd  us  with  ii  dcfinUinn 

a.   .  II  uiciiKiirc  oi  I'orci';  iiiul  tlicv  also  uliow   how  to  coiiipouiui,  and 

ncriit'on'   liow  to  rusolve,  t'orcos ;   ami  also  liow  to   iaveftigati'   tlie 

eoiiditioiis  of  (■(luilibriuin  or  motion  of  a  single  particle  subjected  to 

given  forces. 


169.    Remarks  on  Law  III.— According  to  Law  III,  if  one 

lM)dy  presses  or  draws  another,  it  is  pressed  or  drawn  by  this  other 
with  an  e(|ual  force  in  the  opposite  direction  (Art.  10).  A  horse 
towing  a  Ijoat  on  a  canal,  is  pulled  backwards  l>y  a  force  e(]ual  to  that 
which  he  impresses  on  the  towing  ro|ie  forwards.  If  one  body  strikes 
another  body  and  clmiitres  the  motion  of  the  other  body,  its  own 
motion  will  Ite  changi  in  an  e(iual  quantity  and  in  the  opi)osite 
direction;  for  at  each  instant  during  the  impact  the  l)odio8  exert  on 
eaeJi  other  e<|ual  and  opposite  pressures,  and  the  momentuui  that  one 
body  loses  is  eiputl  to  that  wliicli  the  other  gains. 

Tlie  earth  attracts  a  falling  pebble  with  n  certain  force,  while  the 
peblile  attracts  the  earth  with  un  eijual  force,  'fho  result  is  thai 
wliile  the  pebble  moves  towards  the  earth  on  account  of  its  attrac- 
tion, the  <!artli  also  moves  towards  tlie  pebble  under  the  influence  of 
the  attraction  of  the  latter  ;  but  the  mass  of  the  earth  being  enor- 
mously greater  than  that  of  the  pebble  while  the  forces  on  the  two 
arisinjr  from  their  nnitual  attractions  are  e()ual,  the  motion  jircKluced 
thereby  in  the  earth  is  almost  incoiiiparalily  less  than  that  jiroduced 
in  the  pebble,  and  is  consecpiently  insensible. 

It  follows  that  the  sum  of  the  (lunntities  of  motion  parallel  to  any 
fixed  direction  of  tlie  particles  of  any  system  influencing  one  another 
in  uiiy  possible  way,  remains  unchanged  by  their  mutual  action. 
'I'lieivfori!  if  the  centre  of  gravity  of  any  system  of  niutually 
iiiHuencing  particles  is  in  motion,  it  continues  moving  uniformly  in  a 
straight  line,  unless  in  so  far  iis  the  direction  or  velocity  of  its  motion 
is  changed  by  forces  between  the  (lartich's  and  some  ol/icv  matter  not 
livtinKjiiKj  til  the  .ii/.tt,m  :  also  the  centre  of  gravity  of  any  system  of 
particles  moves  just  as  all  the  matter  of  the  system,  il  concentrated  in 
11  iioint,  would  move  under  the  influence  of  forces  equal  and  parallel 
to   tlie   forces   really  iicting  on    ils    different    parts.      (For    further 


TWO    hi  »S   OF  MOTIOS. 


295 


itoou  lis  a  liio- 
is  the  ./■-(■uiii- 
'  xi'i-lod  (<ii 


'itli  11  drfinitiiin 
compoiiiKl,  and 
inveftifrati-  the 
■le  subjected  to 


Law  III,  if  one 
n  hy  this  otlier 

1<>).  A  horse 
ee  e(|iuil  to  that 
me  itody  striltew 

bculv,  its  own 
n  the  o|)|)osite 
l)0(lies  exert  on 
I'litnni  tiiut  one 

rorco,  while  the 

0  result  is  tiiai 
nt  of  its  attnic- 
lie  influenoi'  of 
til  being  eiior- 
•ces  on  the  two 
otion  |ir(K]ure<l 

1  that  produeed 

[larallel  to  any 
ng  one  another 
mntual  action. 
1  of  mutually 
iinifonnly  in  a 
y  of  its  iniitioti 
hi'v  maltei'  not 
any  Hysteni  of 
onceni  rated  In 
1  and  parallel 
(For    further 


remarks  on   these  laws  see  Tait  and  Steele's  Dynamics  of  a  Particle, 
i'honison  and  Tail's  Nat.  I'hil.,  Pratt's  Mechanics,  etc.) 

170.  Two  Laws  of  Motion  in  the  French  Trea- 
tises.— Newtou's  Liiws  of  motiuii  ai'o  not  adojjU'd  in  tlii" 
iniiifipal  French  treatises :  but  we  find  in  them  two  prin- 
;iple.s  only  as  borrowed  from  experience,  viz.: 

FiKST. — The  Lav  of  Inert ia,  that  a  body,  not  acted 
upon  liy  any  force,  would  ^o  on  for  ever  with  a  uniform 
velocity.     Thi.s  coincides  with  Newton's  First  Law. 

Second. — That  the  /'('/(K!//^  communicated  is  proporfional 
lo  the  force.  The  ."i'coik/  and  l/iird  i^aws  of  Motion  are 
thus  reduced  to  this  second  principle  by  the  French  writers, 
esjiecially  I'oisson  and  Laijlace.* 

171.  Motion  of  a  Particle  under  the  Action  of  an 
Attractive  Force. — -1  particle  mores  under  a  force  of 
attraction  irlii^h  is  in  it.i  line  <f  motion,  and  varies  directly 
as  tJa;  distance  of  tfie  particle  from  the  centre  of  force;  it  is 
required,  to  determine  the  motion. 

The  ])oint  whence  the  inlliu>ncc  of  a  force  emanates  is 
called  the  centre  of  force  ;  and  the  force  is  called  an  attrac- 
tive or  a  re/nilsire  force  according  m  it  attracts  or  repels. 

Let  0  l)e  the  centre  of  force,  P  the       ^, 
positi(m  of  the  particle  at  any  time,  /,  r     ' 
its  velocitv  at  that  time,  and  let  OP  =:  .r. 


-4A 


Fig.BO  MX 


and  OA  =  a,  where  A  is  the  position  <)t  the  particle  when 
'.  =  0  ;  let  /t  =  the  absolute  force  j  that  is,  the  force  of 
attraction  on  a  unit  of  mass  at  a  unit's  distance  from  0, 
which  is  supposed  to  be  known,  and  is  sometimes  called 
the  strcHf/th  of  the  tittraction.     At  pn'sent  we  shall  suppose 


•  ParWinHOti'd  MechnniCK,  p.  187.  800  paper  by  Pr.  Whowoll  mi  the  princlplei* 
«r  nyiiiiniics,  piirliculurly  aa  atated  by  Fruiicb  writtTi*  la  tla'  EkliiibufKb  Juuraul  of 
Scli'uce,  Vol.  VIII. 


290 


A    VAHIAlihK  ATTItAcrrVh:  FORCE. 


tlie  mass  of  the  particle  to  ho  unity,  as  it  simplifies  the 
('(Illations.  'Plien  fix  is  the  magnitude  of  the  force  at  the 
distance  ,»•  on  tlie  particle  of  unit  mass,  or  it  is  the  accelera- 
tion at  P  ;  and  the  e<(uation  of  motion  is 


(IP 


=    —  fix 


(1) 


the  negative  sign  heing  taken  hecause  the  tendency  of  the 
force  IS  to  diminish  x  ; 


'idx  (Px 


dt^ 


=  —  2fj.x  dx. 


Integrating,  we  get 


d^ 
dl^ 


u  (rt2  -  a^), 


(2) 


if  the  particle  be  at  rest  wlien  x  =  a  and  /  =  0, 

=  fiidt, 


the  negative  sign  being  taken,  because  ,r  decreases  as  t 
increases.  Integrating  again  between  the  limits  correspond- 
ing to  /  =  ^  and  ^  =  0, 

COS"'  -  =  ui/, 
a       ^    ' 


t  =  ~T  COS   '  -• 


(3) 


From  (2)  it  appears  that  the  velocity  of  the  particle  is 
zero  when  x  =  a  and  —  a  ;  and  is  a  maximum,  viz.:  tifi^, 
when  ;(•  =  0.  Hence  the  particle  moves  from  rest  at  A:  its 
velocity  increases  until  it  reaches  0  where    it   becomes  a 


B. 


iimplifios  the 

;  force  at  the 

the  accelera- 


(1) 
deiicy  of  the 


(2) 


0, 


creases  as   / 
i  correspond- 


(3) 

he  particle  is 
m,  viz.:  ajt*, 
est  at  A :  its 
t   becomes  a 


A    VARIAIILK  ATTIIAI'TIVE  FORCE. 


297 


iiiiiximum,  and  wliore  tlie  force  is  zero ;  tlie  particle  passes 
tlu'ough  that  point,  and  its  velocity  decreases,  and  at  A',  at 
a  distance  =:  —  n,  becomes  zero.  From  this  point  it  will 
return,  under  tlie  action  of  the  Force,  to  its  original  posi- 
tion, and  continually  oscillate  over  the  space  'Za,  of  which 
0  is  the  middle  point. 

Fi-om  (3)  we  find  when  a;  =  rt,  ^  =  0  and  when  x  =  0, 

/  =  — j-;  so  that  the  time  of  passing  from  A  to  0  =  — v, 
and  the  time  from  0  to  A'  is  tiie  same,  so  that  the  time  of 

7T 


oscillation  from  A  to  A'  is 


I'his  result  is  remarkable, 


as  it  shows  that  the  time  of  oscillation  is  independent  of 
the  velocity  and  distance  of  projection,  and  depends  solely 
on  the  strength  of  the  attraction,  and  is  greater  as  that  is 
less. 

This  problem  includes  the  motion  of  a  particle  within  a 
homogeneous  sphere  of  ordinary  matter  in  a  straight  shaft 
through  the  centre.  For  the  attraction  of  such  a  sphere  on 
a  particle  within  its  bounding  surface  varies  directly  as  the 
distance  from  the  centre  of  the  sphere  (Art.  133a).  If  the 
earth  were  such  a  homogeneous  sphere,  and  if  AOA'  (Fig. 
80)  represented  a  shaft  running  straight  through  its  centre 
from  surface  to  surface,  then,  if  a  jiarticle  were  free  at  one 
end,  A,  it  would  move  to  the  centre  of  the  earth,  0,  where 
its  velocity  would  be  a  maximum,  and  thence  on  to  the 
opposite  side  of  the  earth,  A',  where  it  would  come  to  rest; 
then  it  would  return  through  the  centre,  0,  to  the  side,  A, 
from  where  it  started  ;  and  its  motion  would  continue  to  be 
oscillatory,  and  thus  it  would  move  backwards  and  forwards 
from  one  side  of  the  earth's  surface  to  the  other,  and  the 
time  of  the  oscillation  would  be  independent  of  the  earth's 
radius;  that  is,  at  whatever  point  within  the  earth's  surface 
the  particle  be  placed  it  would  reach  the  centre  in  the 
same  time. 


i^m 


mm^ 


'-J^R  A    rAKlAHLE   HKt'VLStVK  FORCE. 

Cow..— T(i  find  f/iis  fiitic.     Sinco  /t  is  the  attracKon  at  a 
unit  nf  tlistaiico  ami  (j  Hie  attraction  at  the  distance  R,  we 

have  /x  —  •-,  which  in  /  =  --,   elves 


_  ""      III 


for  the  time  it  would  take  a  body  to  move  from  any  point 
within  the  earth's  surface  to  the  centre. 
If  we  put^  z=  32|  feet  and  R  =  3!)(J3  miles  we  get 

/  =  21  m.  f)  s.  about, 

which  would  be  the  time  occupied  in  passing  to  the  earth's 
centre,  however  near  to  it  the  body  might  be  placed,  or 
however  far,  so  long  as  it  is  within  the  surface. 

172.  Motion  of  a  Particle  under  the  Action  of  a 
Variable  Repulsive  Force.— Lei  the  force  be  one  of 
repulsion  and  vary  as  the  distance,  then  this  equation  of 
motion  is 

^  — 
dt»  -  ^'^• 

Tjet  ua  suppose  the  particle  to  be  projected  from  the  cen- 
tre of  force  with  the  velocity  v^ ;  then  we  have 

rf^  =  ''-^  +  V;  (1) 

As  t  increases  x  also  increases,  and  the  particle  recedes 
further  and  further  from  tlie  centre  of  force;  and  the 
velocity  also  increases,  and  ultimately  eqtuils  oo  when  x  = 
t  =  <x>.     Thus  in  this  case  the  motion  is  not  oscillatory. 


1 


A    WMilAlilh:  ATTRACTirK   FORCK. 


attraction  at  a 
listance  II,  we 


om  any  point 
s  we  get 


to  the  earth's 
be  placed,  or 

Action  of  a 

■'cc  he  one  of 
B  equation  of 


from  tlie  cen- 

0 


(1) 


irtiole  recedes 
Tce;  and  the 
30  when  X  = 
oscillatory. 


173.  Motion  of  a  Particle  under  the  Action  of  an 
Attractive  Force  which  is  in  the  line  of  motion,  and 
which  varies  Inversely  as  the  Square  of  the  Distance 
from  the  Centre  of  Force. 

Let  0  (Fig.  80)  bo  the  centre  of  force,  P  tlie  po.sition  of 
tlio  particle  at  the  time  t;  and  A  the  position  at  rest  when 
/  =  0,  so  that  the  particle  starts  from  A  and  moves  to- 
wards O.  Tjet  OP  =  .r.  OA  =  a,  and  fi  =  the  absolute 
force  as  before  or  tbe  acceleration  at  unit  distance  from  O. 
Then  the  equation  of  motion  is 

(Px  _        fJ^ 
Multiplying  by  'idx  and  integrating,  we  get 


dP  =  ^"  L-  -  ah 


(1) 


whicn  gives  the  velocity  of  the  i)article  at  any  distance,  a; 
from  the  origin. 
From  (1)  we  have 


dx  _  /"in 

dt  ~  ~  W    a 


2li  Vnx  —  x^ 


the  negative   sign  being  taken  l)ecaaso  in  the  motion  to- 
wards 0,  X  diminishes  as  I  increases.     This  gives 


V"- 


^^dt=  -^'^ 


\} 


y/ax  —  3? 
a  —  'ix        a         1 


dx. 


* 


300  VELOCITY  1\    FALIJ.xa. 

Integrating  and  taking  the  limits  corrospojitling  \r)t=:t 
and  t  =  0,  we  have 


t 


V^/' 


/ :      a  ,  ''ix       Txa 

y  ax  —  x^  —  -  vers"' 1 — - 

2  a         % 


(2) 


which  gives  the  vahie  of  l. 

When    the   particle  arrives  at  0,  a:  =  0,  therefore  the 
time  of  falling  to  the  centre  O  from  A  is 


From  (1)  we  see  that  the  velocity  =  0  when  x  ■=.  a\  and 
=  cc  when  y  =  0  ;  hence  the  velocity  increases  as  the 
particle  approaches  the  centre  of  force,  and  nltimately, 
when  it  arrives  at  the  centre,  becomes  infinite.  And 
although  at  any  point  very  near  to  0  there  is  a  very  great 
attraction  tending  towai'ds  O,  at  the  point  0  itself  there  is 
no  attraction  at  all ;  therefore  the  particle,  approaching 
the  centre  with  an  indefinitely  great  velocity,  must  pass 
through  it.  Also,  everything  Ijcing  the  same  at  equal 
distances  on  either  side  of  the  centre,  we  see  that  the 
motion  must  be  retarded  as  rapidly  ^s  it  was  accelerated, 
and  therefore  the  particle  will  proceed  to  a  point  A'  at  a 
distance  on  the  other  side  of  O  equal  to  that  from  which  it 
started  ;  and  the  motion  will  continue  oscillatory. 

174.  Velocity  acquired  in  Falling  through  a  Great 
Height  above  the  Earth. — The  i)receding  case  of  motion 
includes  that  of  a  body  falling  from  a  great  height  above 
the  earth's  surface  towards  its  centre,  the  distance  through 
which  it  falls  being  so  great  that  the  variations  of  the  earth's 
attraction  due  to  the  distance  must  be  taken  into  account. 

If  a  sphere  attracts  an  external  particle  with  a  force  which 
varies  inversely  as  the  s(juaro  of  the  <listance  of  the  particle 


(lintr  to  /  =  / 


na 


(2) 


therefore  the 


n  X  =  a;  and 
reuses  as  tlie 
id  ultimately, 
iifiiiite.  And 
s  a  very  great 

itself  there  is 
,  approaching 
ty,  must  pass 
ame  at  ecpial 

see  that  the 
as  uccelerated, 

point  A'  at  a 
from  which  it 
tory. 

mgh  a  Great 

Miso  of  motion 
;  lieight  above 
:ance  through 
J  of  the  earth's 
I  into  account, 
I  a  force  which 
)f  the  piirticle 


VELOriTV  IS  FALLISd. 


301 


from  the  centre  of  the  spliere  (Art.  133a) ;  therefore  if  R  is 
the  earth's  radius,  g  the  kinetic  measure  of  gravity  on  a 
unit  of  mass  at  the  earth's  surface  (Arts.  20,  23),  and  x  the 
distance  of  a  body  from  the  centre  of  the  earth  at  the  time 
/,  then  the  e((uation  of  motion  is 


iPx 


IP 


dp  ~       '-^7?' 

which  is  the  same  as  the  etpiation  in  Art.  173  by  writing  n 
iov  (jU^x  therefore  the  results  of  the  last  Art.  will  apply  to 
this  case.  Substituting  (jW  for  /t  in  (1)  of  Art.  173  we 
have 


(1) 


When  the  body  reaches  the  earth's  surface,  x  =  R  and 

(I)  becomes 

(n  —  R 


.-  =  ...■("  i^')- 


w 


If  a  is  infinite  (2)  becomes 


V  —  y/'igR ', 

so  that  the  velocity  can  never  be  so  great  as  this,  Viowever 
far  the  body  may  fall ;  and  lience  if  it  were  possible  to 
project  a  body  vertically  upwards  with  this  velocity  it  would 
go  on  to  infinity  and  never  stop,  supposing,  of  course,  that 
there  is  no  resisting  medium  nor  other  disturbing  force. 
If  in  (2)  we  put  //  =  32^  feet  and  R  =  31363  miles  we 

V  =  [2-32f  3963 -5280]*  feet  =  6-95  mdes  ; 

80  that  the  greatest  possible  velocity  which  a  body  can 
acquire  in  falling  to  the  earth  is  less  than  7  miles  per 
second,   and  if  a  boily  w>';c  iirojeeted  upwards  with  that 


302  MOTioy  /.v  A  h-KsrsTi.vti  medium. 

velocity,   and    were    to   meet   with    no   resistance  except 
gravity,  it  would  never  return  to  the  earth. 

Cor.— To  find  the  velocity  which  a  body  would  acquire 
in  falling  to  the  earth's  surface  from  a  height  //  above  the 
surface,  we  have  from  (1)  by  putting  x  =-.  R  and  a  =  h-^ R, 

t^  =  2gR4\.  -  --!_)  =  ML. 
\R       R  +  h)        R  +  h 

If  h  be  small  comiiared  with  R,  this  may  be  written 

v^  =  %gh, 

which  agrees  with  (6)  of  Art.  140. 

The  laws  of  force,  enumerated  in  Arts.  171,  173,  are  the 
only  laws  that  are  known  to  exist  in  the  universe  (Pratt's 
Mecbs.,  p.  212). 

175.  Motion  in  a  Resisting  Medium.— In  the  pre- 
ceding discussion  no  account  is  taken  of  (hd  atmospheric 
resistance.  We  shall  now  consider  the  motion  of  a  body 
near  the  surface  of  the  earth,  taking  into  account  the 
resistance  of  the  air,  which  we  may  assume  varies  as  the 
square  of  the  velocity. 

A  particle  tinder  the  action  of  gravitji,  m  a  constant  force, 
moves  in  the  air  supposed  to  be  a  resisting  mcditnn  of 
uniform  density,  of  which  the  resistance  varies  as  the  square 
of  the  velocity  required  to  determine  the  motion. 

Suppose  the  particle  to  descend  towards  the  earth  from 
rest.  Take  the  origin  at  the  starting  point,  let  the  line  i-f 
its  motion  be  the  axis  of  .?• ;  and  let  .r  be  the  distance  <f 
the  particle  from  (he  origin  at  the  time  /,  and  for  con- 
venience let  gk:^  be  the  resistance  of  the  air  on  the  })article 
for  a  unit  of  velocity;  gk^  is  called  the  coefficient  of  rrsisf. 
irncc.     Then  the  resistance  of  the  air  at  the  distance  x  from 


»A. 


stance  except 

ivould  acquire 
t  //  above  the 


ritten 


,  1 73,  are  the 
k^erse  (Pratt's 


-In  tlie  pre- 
atmosphoric 
»n  of  a  body 
account  the 
i'aries  as  the 

nstant  force, 

nieditim  of 

as  the  square 

J  earth  from 
t  the  line  of 
L'  distance  (f 
md  for  eoii- 
the  })article 
nt  (f  rcsisf. 
ance  x  from 


' 


MOTlOy  ly  A    HESISTISG   MEDIUM. 


303 


the  origin  is^pT.y)  ,  which  acts  upwards,  and  the  force  of 
•rravity  is  g  acting  downwards,  the  mass  being  a  unit. 
Hence  the  equation  of  motion  is 

tPx  .  „  Idx-^ 


(1) 


d 


.«.    gdt  = 


dx 
It 


idx\^ 


-KD' 


Integrating,  remembering  that  when  ^  =  0,  «  =  0,  we 


get 


I  dx 
gt  =  ~jr  log  ^^,  (Calculus,  p. -259,  Ex.  5). 


1-k 


dt 


Passing  to  exponentials  we  have 

1  eJcgt  —  e-^ 


dx 
dt 


k  ekgt  +  e-*ff< ' 


(2) 


which  gives  the  velocity  in  terms  of  the  time.     To  find  it  in 
terms  of  the  space,  we  have  from  (1) 


•^^m 


=  2gJ<^dxi 


(3) 


observing  the  proper  limits ; 


304 


MOTlOy  OF  ASCtWT  J.\  rilK  AIR. 


dx^        1 

•••  rf^  =  F^  ^1  -  ^-^*'")'  (4) 

which  gives  the  velocity  in  terms  of  tiie  distance. 

Also,  iiitegratiiig  (2)  taking  the  same  limits  as  ocfore, 
we  get 

gk^x  =  log  (e*?<  +  e->^t)  —  log  3  ; 


...      2f9r*"a:  —  gkgi  ^  ^-kgt^ 


(5) 


which  gives  the  relation  between  the  distance  and  the  time 
of  falling  through  it. 

As  the  time  increases  the  term  <'-*!7<  diminishes  and  from 
(5)  the  space  increases,  becoming  infinite  wiien  the  time  is 
infinite;  but  from  {-i),  as  tiie  time  increases  the  velocity 
becomes  more    nearly   uniform,    and    wiien    t  =  rj. ,    the 

velocity  =  ,  ;  and  althougli  this  state  is  nevei-  reached,  yet 

it  is  that  to  which  the  motion  a]iproaches. 

176.  Motion  of  a  Particle  Ascending  in  the  Air 
against  tb"  Action  of  Gravity.— Let  us  suppose  tlie 
particle  to  .,{■  projected  upwards,  that  is,  in  a  direction 
cdutrary  to  that  of  the  action  of  gravity,  with  a  given 
velocity,  v,  it  is  required  to  determine  the  motion. 

Let  us  supjiose  the  particle  to  be  of  the  same  form  and 
size  as  before,  and  the  same  coefficient  of  resistance. 
Then,  taking  .r  positive  upwards,  both  gravity  and  the 
resistance  of  the  air  tend  to  diminish  the  velocity  as  t 
increases;  so  that  the  equation  of  motion  is 


tP. 


(I) 


(4) 

lice. 

lits  as  ocfore, 

(5) 
and  the  time 


shes  and  from 

n  the  time  is 

i  the  velocity 

f  =  cc ,    the 

r  reached,  yet 


;  in  the  Air 

siil){)ose  tlie 
n  a  direction 
with  a  given 
tion. 

line  form  and 
if  resistance, 
vity  and   tlie 

velocity  as  t 


(I) 


MOTioy  OF  AscK.xr  i.\  riiE  air. 

dx 

=  —  kgdt; 


305 


dk 


dt 


dx\^ 


dx 
.*.    tan~i  k  -r-  •=.  tan""^  (kv)  —  gkl ; 

(Calculus,  J).  :244,  Ex.  3),  since  the  initial  velocity  is  v. 
Taking   tiie   tangent  of  l)oth  members  and  solving  for 
,  we  get 


dx 
di 


dx  _  1      vk  —  tan  kgt 
dt  ~  Tc    1  +  vk  tan  kgt ' 


(2) 


which  gives  the  velocity  in  terms  of  the  time.     To  find  it 
in  terms  of  the  distance,  we  have  from  (1) 


•••    log 


'+-(ir 


r=  -  2^P.r; 


1  +  ^•^«'2 

.• .     {^^  =  u'^e-^k'x  -  ^2  (1  -  e-«!7*'*), 
which  gives  the  velocity  in  terms  of  the  distance. 


(3) 
(4) 


Also,  integrating  (2)  after  substituting  sine  and  cosine 
for  tangent,  and  taking  the  same  limits  as  before,  we  get 


yL^x  =  log  {vk  sin  kyt  +  cos  kgt) ; 


(5) 


which  gives  the  space  described  by  the  particle  in  terms  of 
the  time. 


300 


MOTION  OF  ASCEST  IX  THE  AIR. 


Cor.  1.— To  liiul  the  gmitt'st  height  to  which  the  par- 

dx 
tide  will  ascend  put  the  velocity,  ^'^  =  0,  in  (3)  and  get 


.  =  .^^,log(l+iV), 

which  is  the  distance  of  the  highest  point 
Putting  1^  =  0  in  C^)  we  get 

/  =  ■=—  tan~*  vk, 
kg 


(6) 


(7) 


which  is  the  time  required  for  the  particle  to  reach  the 

highest  point.     Having   reached    the  greatest   height,  the 

])article  will  begin   to  fall,  and  the  circumstances  of  the 
fall  \vill  be  given  by  tlie  e(iuations  of  Art.  175. 

Cou.  2.— Since  k  is  the  same  in  this  and  Art.  175,  we 
iniiv  compare  the  velocity  of  projection,  r,  with  that  Avhich 
the  particle  would  acquire  in  descending  to  tlie  point 
whence  it  was  projected.  Denote  by  t\  the  velocity  of 
tlie  ])article  when  it  reaches  the  point  of  starting.  From 
(3)  of  Art.  175  we  have 


W  '"^  r 


Pj'„2' 


and   placing  this  value  of  x  cciual  to  that  given  in  ((!). 
we  get, 


r--xv  = '  +  **"•■ 


uliiih   is  less  tiian  r;    hence   the  velocity  acquired  in  the 


MOTlOy   OF  A    PROJECTILE. 


307 


:h  tilt'  par- 
and  get 


(7) 

)  reach  tlie 
lieiglit,  the 
iR'os  of  the 


Lrt.  175,  wc 

f'at  which 

tlie   ])oint 

velocity  of 

iiig.     From 


ven  ill  (()). 


descent  is  less  than  that  lost  in  the  ascent,  as  might  have 
been  inferred. 

Cor.  3. — Substituting  (0)  in  (5)  of  Art.  175,  we  get  for 
the  time  of  the  descent, 


ired  in  the 


t  =  ^-log(^/i  +  kh'i  +  kv), 

which  is  dilTerent  from  the  time  of  the  ascent  as  given  in 
(7).  (.See  Price's  Anal.  Mech's,  Vol.  I,  p.  lOG;  Ventnroli's 
Mech's,  p.  82  ;  Tait  and  Steele's  Dynamics  of  a  Particle, 
p.  237. ) 

177.  Motion  of  a  Projectile  in  a  Resisting  Me- 
dium.— 'I'he  theory  of  the  motion  of  i)rojectiles  in  vacuo, 
wiiich  was  examined  under  the  head  of  Kinematics,  affords 
results  which  (litfor  greatly  from  those  obtained  by  direct 
experiment  in  the  atmosphere.  When  j)rojectiles  move 
with  but  .small  velocity,  the  discrepancy  between  the  para- 
bolic theory,  and  what  is  found  to  occur  in  practice,  is 
small  ;  but  with  increasing  velocities,  as  those  with  which 
l)alls  and  siiells  traverse  their  paths,  the  air's  resistance 
increases  in  a  higher  ratio  tha';.  'he  velocity,  so  that  the 
diserei)ancy  becomes  very  great. 

The  most  ..nportant  application  of  the  theory  of  lu'ojec- 
tilcs,  is  that  of  (Junnery,  in  which  the  motion  takes  place 
in  the  air.  If  it  were  ailowal)le  to  neglect  the  resistance  of 
the  air  the  investigations  in  Part  II  would  explain  the 
theory  of  gunnery  ;  but  when  the  velocity  is  consideral)le, 
the  atmospheric  resistance  changes  the  nature  of  the  tra- 
jectory 80  much  as  to  render  the  c(»iiclusions  drawn  from 
tlie  meory  of  projectiles  in  vacuo  almost  entirely  inap- 
])li('able  in  jn'iictice. 

Tile  problem  of  gunnery  may  be  stated  as  follows: 
(liven  a  projectile  (tf  ktiowii  wciglil  and  dimeesions, 
starling  witii  a  known  velocity  at  ii  known  angle  'if  eleva- 


308  MOTION  OF  A    PROJECTILE. 

tioii  ill  a  calm  iitmosj)Iierc'  of  Jii)i)i-oximiitoly  known  duii^it  v ; 
to  find  its  range,  time  of  flight,  velocity,  direction,  and 
position,  at  any  !nomeiif ;  or,  in  other  words,  to  construct 
its  trajeefory.  Tiiis  jiroljlem  is  not  yet,  however,  suscepti- 
l)le  of  rigorous  treatment ;  mathematics  has  hitherto  proved 
uiii'ble  to  furnisli  comi)!etc  formula}  satisfying  the  condi- 
tions. The  resistance  of  the  air  to  slow  movements,  say  of 
10  feet  per  second,  seems  to  vary  w^ith  V\q  first  power  of 
the  velocity.  Above  tiiis  the  ratio  increases,  and  as  in  the 
case  of  the  wind,  is  usually  reckoned  to  vary  as  the  scpiare 
of  the  velocity;  beyond  this  it  increases  still  further,  till  at 
1200  feet  per  second  the  resistance  is  found  to  vary  as  the 
cube  of  the  velocity.  The  ratio  of  increase  after  this  point 
is  i)assed  is  supposed  to  diminish  again  ;  but  thoroughly 
satisfactory  data  for  its  deteriniiiation  do  not  exist. 

From  experiments*  made  to  delerinine  the  motion  of 
cannon-balls,  it  appears  that  when  the  initial  velocity  is 
consrderable,  the  resistance  of  the  air  is  more  than  "v'O  times 
lu*  great  as  the  weight  of  the  ball,  and  the  horizontal  range 
is  often  a  small  fraction  of  that  wliich  the  theory  of  pro- 
jectiles in  vacuo  gives,  go  that  the  form  of  the  trajectory  is 
very  different  from  tiiat  of  a  paraliolic  ])a(h.  Such  experi- 
ments have  been  made  with  great  care,  and  siiow  how  littlf 
the  parabolic  theory  is  (o  be  dejiciided  upon  in  determining 
the  motion  of  military  projectiles. 

178.  Motion  of  a  Projectile  in  the  Atmosphere 
Supposing  its  Resistance  to  vary  as  the  Square  of 
the  Velocity. — .1  pari iclc  under  f/i,' (ir/ion  of  (fnirilij  is 
prnjvr.h'd  /roiii  a  t/ircii  point  iu  n  (/ireii  dirrrfioii  irilli  n 
(fimi  rclorifi/,  and  mores  in  the  atmosphere  ir/iosr  rcsistinice 
is  assumed  to  rari/ as  the  st/iiare  of  the  veloaift/ ;  to  dvler- 
iiiiiie  the  niolioii. 


*  Sen  KncyclopiwUa  Biitnniiira,  Art.   Ouuiiory  ;    ulxu   Itobiii's  Gnmiiiv,  aiul 
IIuttou'H  TraclH, 


wii  (k'ii^it  V ; 
'ectioii,  ami 
0  coiistnict 
T,  suscopti- 
lorto  proved 

the  condi- 
eiits,  SUV  of 
sfc  power  of 
d  as  ill  the 

tlic  square 
thcr,  till  at 
vary  as  the 

•  this  point 
thoroughly 
ist. 

motion  of 
velocity  is 
111  '^0  times 
)ntal  range 
ory  of  pro- 
rajct'tory  is 
U'h  experi- 

•  how  littlf 
etermining 


mosphere 
Square  of 

grttrilji  is 
ion  irilh  " 
•  n'sintinnc 

:  lit  (li'li'v- 


UiiiuuTv.  ami 


MOTION  OF  A    VKOJUCTILE. 


309 


Take  the  given  point  as  origin,  the  axis  of  x  horizontal, 
the  axis  of  j/  vertical  and  positive  upwards,  so  that  the 
direction  of  projection  may  be  in  the  plane  of  ay.  Let  r 
lie  the  velocity  of  projection,  (j  tiie  acceleration  of  gravity, 
K  the  angle  between  the  axis  of  x  and  the  line  of  projection, 
and  let  the  resistance  of  the  air  on  the  particle  be  k  for  a 
unit  of  velocity;  then  the  resistance,  at  any  time,  t,  in  the 

j\  ;  and  the  x-  and  w-ooTriponents  of 
this  resistance  are,  respectively, 


dx 
Tt' 


and    k 


ds 
Jt' 


dt' 


Then  the  equations  of  motion  tire,  resolving  horizontally 

and  vertically, 

d^x  _       ,  dfi    dx  /.v 

dp  -  ~    dl    dV  ^  ' 


iPy  _ 


ds    dy 


d(^ 


-  -  -9-^ at  dt 


(2) 


From  (1)  we  have 
(dx\ 


d 


idx\ 
\dt) 


dx 

=  —  kds ;     .  • .    log 

dx  "^  t;  cos  « 

di 

r.    dx 
since  when  /  =  0,  -jr  =  v  cos  «; 


=  —ks; 


dx 

di 


,-  =  V  cos  «  e 


-k*. 


(■"•) 


Multiplying  (I)  and  (2)  by  dij  and  dx,  respectively,  and 
subtracting  the  former  from  the  latter  we  have 


(Py  dx  -jPxdy  _ 


(4) 


* 


MB.. 


310  MOTIOX   OF  A   PliOJECrrLE. 

Substituting  in  (4)  f  jP  dP  its  value  from  (3)  we  get 

dx^  dx  r'cos^a  *       ^' ' 

Substituting  in  the  second  member  of  (5)  for  dx  its  value 
rfs_^  Wl  +  ^•^.^,  we  get 

du 
Put  -,-  =  p,  and  (6)  becomes 


dx 


( 1  +  P^)^  dp  =  —  -»" ,—  e^  ds. 

^  -f    '        /  ^2  cos'  « 


Integrating,  and  rem;5nbering  tliat  when  s  =  0,  j»  =  tan  «, 
we  get 

p{i  +;>')*  + log  [;>  + (1  +F')*] 

=  c  -  ^-/  -Y    e*^-  (V) 

kv'  COS'*  « 

where  <•  is  the  constant  of  integration  whose  value 

=  tan  «  sec  rt  +  log  (tan  n  -\-  sec  «)  +  -rir^—^-  •    (8) 
From  (5)  we  have 

i'*  cos^ «  dx  ^dxl ' 

whicli  in  (7)  gives 

;;(!  +  ;;»)*  +  log  [p  +  {\  +  ;>«)*]  -  c  =  ^.^|, 


;,(1  +;;!')*  ^  log[/M-  (1  +/''')^]  -c 


kdx.      (!») 


get 

f^dx.       (5) 
Ix  its  value 

(6) 


,p  =  tan 


«> 


(7) 


uo 
COS'*  a        ' 


I  dp 
''  kdx' 

kdjc,      (9) 


MOTlOy   OF  A    PROJECTILE, 

311 
=  A%.    (10) 

*""    P  (1  +i^)*  +  log  [i'  +  (1  +  Z'^)*]  -  c 
From  (4)  we  have 

dx  •  dp  ~  —  (/fi^<*. 

Substituting  this  vahie  of  dx  in  (9)  and  solving  for  dl  we 
get 

^iL =  (i^)irf^.  (11) 

{,.  _  p  (1+ ;^)i  _  log  [;;  +  (1+/)*]  i* 

the  negative  sign  of  dp  being  taken  because  jo  is  a  decreas- 
ing function  of  /, 

Ileplacing  the  value  of  p  =  -•  ,  (9),  (10),  luul  (1 1)  become 


d 


dx  ss 


dx 


*l('+;£)*-M^-('+g)V 


(A) 


dy  = 


dx     dx 


*i('+g)^-[2-('+^r] 


(B) 


-d 


dt  = 


(ty 
dx 


.TfTTT.  (C) 


<^''^-t('^•;;^*-H2-^(■+l:r]s' 


from  which  etiuations.  were  it  possible  to  integrate  then, 

:r,  y,  and  /  might  he  found  in  lerm.s  of   ;   ;  and  if    ;    were 
'  -"  '"  dx  dx 

eliminated  from  the  two  integrals,  of  (A)  and  (H),  the  re- 
sulting equation  in  terms  of  .«'  and  y  would  bo  that  of  the 


3li 


MOTIOX   OF  A    rnOJECTlLE. 


rc([aired  trajectory.  But  those  equations  cannot  be  inte- 
grated in  finite  terms;  only  approximate  solutions  of  them 
can  be  made  ;  and  by  means  of  these  the  i)ath  of  the  pro- 
jectile may  be  constructed  approximately.  (See  Venturoli's 
Mech.5.,  p.  92,) 

S<iu,aring  (A)  and  (B),  and  dividing  their  sum  by  the 
square  of  (0)  we  get 


2(1 +W-'-[l +('+:!)*] 


(t>) 


which  gives  the  velocity  in  terms  of  -' 


dy 


179.  Motion  of  a  Projectile  in  the  Atmosphere 
under  a  small  Angle  of  Elevation. — The  case  fro- 
(piently  occurs  in  practice  where  tiie  angle  of  projection  is 
very  small,  and  where  the  projectile  rises  but  a  very  little 
above  the  horizontal  line.  In  this  case  the  ecpiation  of  the 
part  of  the  trajectory  that  lies  above  the  horizontal  line 
nuiy  easily  l)e  found ;   for,   the  angle  of  projection  being 

very  small,    . ,  will  Ije  very  small,  and  therefore,  throughout 

the   path   on    the  upi)er  side  of  the   axis  of  x,  powers  of 

'  •    higher  than  the  first  may  l)e  neglected.     In  this  case 

then 

rfs  =  (?a; ;       .  • .    s  =:  x\ 

wliich  in  (5)  of  Art.  178,  becomes 


(7-^  _ 
dx 


!l 


V*  cos*  « 


^^dx ; 


lot  be  in  to- 
ons of  them 
of  the  pro- 
!  Venturoli's 

sum  by  the 


(1)) 


tmosphere 

le  case  fro- 
)ruj(.'ction  is 
a  very  little 
at  ion  of  the 
■izontal  line 
potion  being 

,  throughout 

r,  powers  of 

In  this  case 


EXAMPLES. 


313 


Integrating,  we  get 
dy 


dx 


—  tan  «  =  —  :y77 


2^•t•2  cos' « 


(e"»'-l); 


dy 


sinjc  when  a;  =  0,  "j'  =  tan  «. 


Integrating  again  we  get 

gx 
y  =  X  tan  «  +  ^:^;r^^«  -  4^.2^  eos'  a 

Expanding  e^**  in  a  series,  (1)  becomes 


-.7:oA^(^-l)-(l) 


gi? 


gks? 


y  =  x  tan  «  -  .^^^^l „  -  3^2  pos^  « 


(2) 


the  first  two  terms  of  which  represent  the  trajectory  in 
vacuo.     [Sce(:5)of  Art.  151.] 
From  (3)  of  Art.  178,  we  have 


dt  = •  dx 

VCO8  a 

e^-1 

hv cos  rt 


t 


(3) 


which  gives  the  time  of  flight  in  terms  of  the  abscissa. 

The  most  comjjlete  and  valuable  series  of  experiments 
on  the  motion  of  projectiles  in  the  atmosphere  that  has  yet 
been  maxle,  is  that  of  Prof.  F.  Bashforth  at  Woolwich. 


EXAMPLES. 


1.  Find  how  far  a  force  equal  to  the  weight  of  n  Hts., 
would  move  a  weight  of  m  lbs.  in  t  seconds ;  and  find  th« 
velocity  acquired. 


^m 


311 


EXAMl'LKS. 


Here  P  =  n,  and   W  =  in ;  therefore  from  (1)  of  Art 

25  we  have 

ng 


n  =  — / ; 


f 


m 


which    hi   (i)  and   (5)   respectively   of    (Art.    D),    gives 
v  =  —  ;  and  s  =  i  — i^ 


m 


m 


lbs. 


2.  A  body  weighing  n  lbs.  is  moved  by  a  constant  force 
which  generates  in  the  body  in  one  second  a  velocity  of  a 
feet  per  second  :  find  the  force  in  pounds.       .        na 

9 

3.  Find  in  what  time  a  force  of  4  lbs.  would  move  a 
weight  of  9  lbs.  through  49  ft.  along  a  smooth  horizontal 
plane ;  and  And  the  velocity  acquired. 

Ans.  t  =  --=; ;  V  =  ^t. 
V2g 

4.  Find  the  number  of  inches  through  which  a  force  of 
one  ounce,  constantly  exerted,  will  move  a  mass  weighing 
one  lb.  in  half  a  second.  A7is.  3g  (J)'. 

5.  Two  weights,  P  and  Q,  are  connected  by  a  string 
which  passes  over  a  smooth  peg  or  pulley  ;  required  to 
determine  the  motion. 

Since  the  peg  or  pulley  is  perfectly 
smooth  the  tension  of  the  string  is  the 
same  throughout;  hence  the  force  which 
causes  the  motion  is  the  difference  between 
the  weights,  P  and  Q,  the  weight  of  the 
string  l)cing  neglected.  The  moving  force 
therefore  is  P  —  Q\  but  the  weight  of  the 
maj^s  moved  is  /'  +  Q.  Hence  substituting 
in  (1)  of  Art.  25,  we  get 

P+Q 


Fig.  80'i. 


Q  = 


/; 


(1)  of  Art 


y),     gives 


nstant  force 

velocity  of  n 

na  ,, 
IS.  —  lbs. 
9 

»nld  move  a 
h  horizontal 


;  V  =  ^t. 

ch  a  force  of 
iiss  weighing 

by  a  string 
required   to 


f\ 


p 


1 


Fig.  80rt. 


KXAMl'LES. 

P  —  Q 

which  is  tlic  acceleratioti. 

Substituting  this  in  (4)  and  (5)  of  Art.  'J,  we  have 


315 
(1) 


(2) 
(3) 


which  gives  the  velocity  and  space  at  the  time  /,  the  initial 
velocity  i\  being  0. 

0.  A  bodv  whose  weight  is  Q,  rests  on  a  smooth  hori- 
zontal tableland  is  drawn  along  by  a  weight  /^attached  to 
it  by  a  string  passing  over  a  pulley  at  the  edge  of  the  table; 
find  the  motion  of  the  bodies. 

Sit>ce  the  weight  Q  is  entirely  supported  by  the  resistance 
of  the  table,  the  moving  force  is  the  weight  P,  hanging 
vertically  downwards,  and  the  weight  of  the  mass  moved  is 
P  +  Q;  therefore  from  (1)  we  have 


/  = 


P  +  Q 


(1) 


an<l  this  in  (4)  and  (.5)  of  Art.  9   gives   the  velocity  and 
space. 

7.  Required  the  tension,  T,  of  the  string  in  the  pre- 
ceding example. 

Here  the  tension  is  evidently  tliat  force  whicli.  acting 
along  the  string  on  the  body  whose  weight  is   Q.  produces 

,n  it  the  acceleration,  75^^.'/.  «'hI  therefore  is  measured 

by  the  mass  of  Q  into  its  acceleration.     Hence 


316 


EXAMPLES. 


0 


9  =  -n 


PQ 


P+Q"      P+Q 


8,  Find  the  tension,  T.  of  the  string  in  Ex.  5. 

Here  the  tension  eoiuls  ilie  weight  Q,  phis  tlie  foree 
which,  acting  along  the  string  on  Q,  produces  in  it  the 
acceleration 

P+Q^' 


T=  Q  + 


Q  P 


u   P+Q"" 


2PQ 
~  P+Q' 

or  it  equals  P  minus  the  accelerating  force  which,  of  course, 
gives  the  same  result. 

9.  Two  weights  of  9  lbs.  and  7  lbs.  hang  over  a  i)ullev,  as 
in  Ex.  5  ;  motion  continues  for  5  sees.,  when  the  string 
breaks:  find  the  height  to  which  the  lighter  weight  will 
rise  after  the  breakage. 

Substituting  in  {'i)  of  Kx.  5  we  have 

V  =  -^^  ;W  .  5  =  20 ; 

therefore  each  weight  has  a  velocity  of  20  feet,  when  the 
string  breaks.  Hence  from  (0)  of  Art  fl,  we  have  (calling 
ff  32  ft.) 

.V  =  v/  =  H ; 

that  is.  the  lighter  weight  will  rise  GJ  feet  before  it  begins 
to  descend. 

10.  A  steam  engine  is  moving  on  a  horizontal  plane  at 
the  rate  of  30  miles  an  hour  when  the  steam  is  turned  off; 
supposing  the  resistance  of  friction  to  be  4^5^  of  the  weight, 
lind  how  long  and  how  far  the  engine  will  run  before  it 
stops. 


the   force 
ces  iu  it  the 


ich,  of  cotirse, 


T  a  j)iilloy,  as 
)n  the  string 
r  weight  will 


pet,  when  the 
have  (calling 


'ore  it  begins 

ntal  plane  at 
3  turned  off  ; 
f  the  weight, 
run  before  it 


h:XA.vi'Lj-:s. 


'W: 


Let  ir  be  the  weight  of  tin- enginf;  then  the  resistance 
A  friction  is  t,\a.  and  this  is  directly  opposed  to  motion, 


W  _  W 
400  ~  g-'  ' 


•••   /- 


400  ■ 


„,,         ,     .,  .    .,„      .,  ,  30x1760x3 

I  he  velocity,  r,  is  30  miles  an  hour  =  — — —  ---     =  44 
'  60  X  60 

feet  per  second.     Substituting  these  values  of/ and  v  in  the 

equation  v  =  ft,  we  get 

44  =  ;^;; 
.     t  =  550  sees., 

which  is  the  time  it  will  take  to  bring  the  engine  to  rest  if 
the  velocity  l)e  retarded  ^^j-  feet  per  second. 
Also  v^  =  'ifs,  therefore 

s  =  4Aiii^-4jio  _  12100  feet. 

11.  A  man  whose  weight  is  11',  stands  on  the  platform 
of  an  elevator,  as  it  descends  a  vertical  shaft  with  a  uniform 
acceleration  of  4 (/;  find  the  pressure  of  the  man  upon  the 
platfcrm. 

Let  P  be  the  jtressure  of  the  man  on  the  platform  when 
it  is  moving  with  an  acceleration  of  ^/  ;  then  the  moving 
force  is  W  —  P ;  and  the  weight  nii)ved  is  W;  therefore 


w-P=^^hr, 


p  =  ^w. 


12.  A  plane  supporting  a  eight  of  12  ozs.  is  descending 
with  a  uniform  acceleration  of  10  ft.  ])cr  second  ;  find  the 
pressure  that  the  weight  exerts  on  the  plane. 

Ans.  8 J  ozs. 


:{1H  EXAMPLEH. 

i:}.  A   weight  of  24  lbs.  hanging  over  the  edge   of  a 
smooth    table  drags  a  weight  of    Vl  His.  along  the  table; 
tiiid  (1)  the  accelei-ation.  and  (--i)  the  tension  of  the  string. 
Am.   (1)  ai^  ft.  per  sec;    (^)  8  lbs. 

14.  A  weight  of  8  lbs.  rests  on  a  platform  ;  find 
its  pressure  on  the  phitforni  (1)  if  the  latter  is  de- 
scending with  an  aeeeleration  of  \(U  and  (2)  if  it  is 
aseeiuling  with  the  same  acceleration. 

Am.   (1)  7  ll)s.;  (2)  9  lbs. 

15.  Two  weights  of  80  and  70  lbs.  hang  over  a  smooth 
pulley  as  in  Ex.  .5  ;  find  the  space  through  which  they  will 
move  from  rest  in  3  sees.  Aiis.  9|  ft. 

16.  '•^wo  weights  of  15  and  17  ounces  respectively  hang 
over  a  smooth  pulley  as  in  Ex.  5  ;  find  the  space  de- 
scribed and  the  velocity  acquired  in  five  seconds  from  rest. 

Am.  ,s  =  25,  V  —  10. 

17.  Two  weights  of  5  lbs.  and  4  lbs.  together  pull  one 
of  7  lbs.  over  a  smooth  fixed  pulley,  by  means  of  a  con- 
necting string;  and  after  descending  through  a  given 
space  the  4  lbs.  weight  is  detached  and  taken  away  without 
interrnpting  the  motion  ;  find  through  what  space  the 
remaining  5  lbs.  weight  will  descend. 

Ann.  Through  |  of  the  given  space. 

18.  Two  weights  are  attached  to  the  extremities  of  a 
string  which  ia  hung  over  a  smooth  pulley,  and  the  weights 
are  observed  to  move  through  0.4  feet  in  one  second  :  the 
motion  is  then  stopped,  and  a  weight  of  5  lbs.  is  added 
to  the  smaller  weight,  which  then  descends  through  the 
same  space  as  it  ascended  before  in  the  same  time  ;  deter- 
mine the  original  weights.  Am.  %  lbs.;  ^  lbs. 

19.  Find  what  weight  must  be  added  to  the  smaller 
weight  in  Ex.  5,  so  that  the  acceleration  of  the  system  may 


e  edge  of  a 

If  the  table  ; 

f  tlie  string. 

(i)  8  ll)s. 

tform  ;  tiiul 
atter  is  de- 
(2)    if    it   is 

(2)  9  lbs. 

\er  a  sniootli 
ich  tliey  will 
i/is.  9|  ft. 

actively  hang 
he  space  de- 
is  from  rest. 
5,  V  =  10. 

her  pull  one 
ins  of  a  con- 
iigli  a  given 
fiway  without 
at   space  the 

;iven  space. 

iremities  of  a 

d  the  weights 

}  second  :  the 

lbs.  is  added 

through  the 

time  ;  deter- 

lis. ;  V-  l''s- 

I  the  smaller 
le  system  nuiy 


KXAMl'LKS. 


319 


have  the  same  numerical  value  as  before,  but  may  l)e  in 


ihe  opi»osite  direction. 


A  US. 


20.  A  body  is  projected  up  a  rough  inclined  plane  with 
tiie  velocity  which  would  be  acquired  in  falling  freely 
through  12  feet,  and  just  reaches  the  top  of  the  plane; 
the  inclination  of  the  i»'",ne  to  the  horizon  is  00",  and  the 
coet!icient  of  friction  is  equal  to  tan  30°;  find  the  height  of 
the  plane.  Ans.  9  feet. 

21.  A  body  is  projected  up  a  rough  inclined  plane  with 
the  velocity  Ig  ;  the  inclination  of  the  plane  to  the  horizon 
is  30°,  and  the  coefficient  of  friction  is  e([ual  to  tan  15° ; 
lind  the  distance  tilong  the  plane  which  the  body  will 
describe.  Jns.  </ (V3  +  1). 

22.  A  body  is  projected  up  a  rough  inclined  plane ;  the 
inclination  of  the  plane  to  the  horizon  is  «,  and  the  coef- 
ficient of  friction  is  tan  e ;  if  m  be  the  time  of  ascending, 
and  n  the  time  of  descending,  show  that 

(>n\^_  sin  (fc  —  e) 
\»  /  ~  sin  (rt  +  e) 

23.  A  weight  P  is  drawn  up  a  smooth  plane  inclined  at 
an  angle  of  30°  to  the  horizon,  by  means  of  a  weight  Q 
which  descends  vertically,  the  weights  being  connected  by 
a  string  passing  over  a  small  jjullcy  at  the  top  of  the  plane ; 
if  the  acceleration  be  one-fourth  of  that  of  a  body  falhng 
freely,  find  the  ratio  of  Q  to  P.  Ans.  Q  —  P. 

24.  Two  weights  P  and  Q  are  connected  by  a  string, 
and  Q  hanging  over  the  top  of  a  smooth  plane  inclined  at 
30°  to  the  horizon,  can  draw  P  up  the  length  of  the  plane 
in  just  half  the  time  that  P  would  take  to  draw  up  Q ; 
show  that  Q  is  half  as  heavy  again  as  P. 


ite 


320 


KXAMl'LES. 


25.  A  particle  moves  in  a  stniiglit  line  under  the  action 
of  an  attraction  varying  inversely  as  the  (|)th  power  of 
the  distance  ;  show  that  the  velocity  acquired  hy  falling 
from  an  infinite  distance  to  a  distance  a  from  the  centre  is 
equal  to  tlie  velocity  which  would  be  acquired  in  moving 

from  rest  at  a  distance  a  to  a  distance  -:> 


T  the  action 

t!)   {lower  of 

d  I)}'  fiUliiig 

le  centre  is 

in  moving 


CHAPTER    II. 

CENTRAL    FORCES.* 

180.  Definitions. — A  central  force  is  o!ie  which  acts 
directly  towards  or  from  a  fixed  jioint,  urv.l  is  called  an' 
attraclire  or  a  repuhire  force  according  as  its  action  on 
any  particle  is  altrncti">i  or  repulsion.  Tiie  fixed  point  is 
called  tb.o  Centre.  Tiie  intensity  of  the  force  on  any  jiar- 
ticli  is  some  function  of  its  distance  from  the  centre. 
Since  tiie  case  oi  attraction  is  t'lc  most  important  applica- 
tion of  tlie  subject,  we  shall  tal-^  that  as  our  standard  ease; 
but  it  will  be  seen  that  a  simple  change  of  sign  will  adapt 
our  general  formula)  to  repulsion.  If  the  centre  be  itself 
in  motion,  we  may  treat  it  as  fixed,  in  which  case  I  he  term 
"actual  motion  "  of  any  particle  means  its  motion  "rela- 
tive "  to  the  centre,  taken  as  fixed. 

Tlie  line  from  the  centre  to  the  particle,  is  called  a 
Radius  Vector.  The  path  of  the  particle  under  tlie  action 
jf  an  attraction  or  repulsion  directed  to  the  centre  is 
"ailed  its  Orbit.\  All  the  forces  of  nature  with  which  we 
are  acquainted,  are  central  forces;  for  this  reason,  and  lie- 
cause  t.ie  motion  of  liodies  under  the  action  of  central 
forces  is  a  branch  of  the  general  theory  of  Astronomy,  we 
shall  devote  this  chapter  to  the  consideration  of  their 
action. 

181.  A  Particle  under  the  Action  of  a  Central 
Attraction ;  Required  the  Polar  Equation  of  the 
Path. — The  motion  will  clearly  take  place  in  the  plane 
{lassiiig  through  the  centre,  and  the  line  along  which  the 

*  This  chapter  contalni*  the  flrxt  prliiolplc"  of  Miitheinutlcal  Antronomy.     It 
may,  howcvor,  be  oiiiillcd  by  tli«  hHuIciiI  of  KiiKliic«iiii){- 
t  Called  Central  OrbltH, 


»^i 


322 


r'A'.Vr/i"  A 1,    A  TTh'A  <  TIOX, 


particle  is  initially  inojootcd,  as  tliciv  is  iiotliinp  to  With- 
(haw  the  particlo  from  il.  lid  tlio  cc'iilre  of  al  trad  ion,  O, 
\w  tlio  origin,  ami  OX.  OY.  anv 
two  lini's  tiirongh  Oat  right  angles 
to  etich  other,  l)e  the  axes  of  co- 
ortlinates.  Let  (.c.  //)  he  the 
pioition  of  the  partiele  M  at  the 
.time  /,  and  (r,  0)  its  position 
referred  to  polar  co-ordinates. 
OX  heing  the  initial  line.     Then,  \/  p.   g, 

calling   /*   the  central    attractive 
force,  we  have  for  the  components  parallel  to  the  axes  of  x 

and  y,  respectively,  —  P^,  —  P-{,  the  forces  heing  nega- 
tive, since  they  tend  to  diminish  the  co-ordinates.  There- 
fore the  e(iuations  of  motion  are 


(Px  _        pX 


im  ~  r 


(1) 


Multiplying   the   former  hy   y.  and   the  latter  by  x,  and 
subtracting,  v.  e  have 


(Pu  iPx        - 


Integrating  we  have 


dt 


dx 

y-dt 


hi 


(2) 


(8) 


where  h  is  an  undetermined  constant. 
Since  a;  =  r  cos  0,  and  y  —  r  sin  0,  we  have 
dx  =  cos  0  dr  —  r  sin  0  dd, 
dy  =  sin  B  dr  +  r  om  d  dO, 

whicli  in  (3)  gives 


(4) 


C^) 


(3) 


(4) 


CJCXTR.  I ;.   A  TTJi'A  I  'TIOS. 


Agiiin,  multiplying  the  first  and  socond  of  (1)  by  -2(lr 
and  'idy  respci-tively,  and  adding,  we  get 


2dx  ifir  +  -idyiPy  _  _  'iP{.edx  +  rj  dij) 
dP  '  r 


(dj^      dy\  _       , 


2Pdr. 


(6) 


Substituting  in  (0)  the  values  of  dr^  and  <//  from  (4),  we 
have 


(7) 


Put  r  =  '  ;  and  .-.  dr  =  -  '-"„;  and  (T)  becomes 
u  « 

IKTforniiiig   the  differentiation    of  the   first  member,  and 
dividing  by  'idii,  aiid  transposing,  we  get 


tPu   ,  /'         o 


(8) 


which  is  the  diffi'rontinl  fqiiafinn  of  the  orbit  described; 
and  as,  in  any  partienlar  instance,  the  foree  P  will  »)e  given 
in  terms  of  r,  and  therefore  in  terms  of  «,  the  integral  of 
this  ecpiation  will  be  tiie  polar  equation  of  the  re(|uired 
path. 

Solving  (8)  for  /'  we  have 


324  CENTRAL   ATfUACTtON. 

Idhi 


"""C-")^  (») 


which  is  the  same  result  that  was  found  by  a  different  pro- 
cess in  Art.  1G3  for  the  acceleration  along  the  radius 
vector. 

(!oR.  1.— The  general  integrals  of  (1)  will  contain  four 
arbitrary  constants.  One,  //,  that  was  introduced  in  (a), 
and  two  more  will  be  introduced  l)y  the  integration  of  (8). 
If  the  value  of  r  in  terms  of  it,  deduced  from  the  integral 
of  (8),  be  substituted  in  (5).  and  that  e(iuation  be  then 
integrated,  the  fourth  constant  will  be  introduced,  and  the 
path  of  the  particle  and  its  position  at  any  time  will  be 
obtained.  The  four  constant?  must  be  determined  from 
the  initial  circumstances  of  motion ;  viz..  the  initial 
position  of  the  particle,  depending  on  two  independent 
30-ordinatcs,  its  initial  velocity,  and  its  direction  of  pro- 
jcction. 

Cor.  2.— By  means  of  (9)  we  may  ascertain  the  law  of 
the  force  which  must  act  upon  a  particle  to  cause  it  to 
describe  a  given  curve.  To  effect  liiis  we  must  determine 
the  relation  between  «  and  0  from  the  polar  ciiuation  of  the 
orbit  referred  to  the  re(|uired  centre  as  pole  ;  we  iiuist  then 
differentiate  u  twice  with  resjiect  to  0,  and  sul)stitute  the 
result  in  the  expression  for  /',  eliminating  W,  if  it  occurs, 
by  means  of  tlu;  relation  between  n  and  0.  In  this  way  wi. 
siiall  obtain  7'  in  terms  of  u  alone,  and  therefore  of  ? 
alone. 

Coil.  3.— When  we  know  the  relation  between  r  and  « 
from  (0),  we  may  by  (5)  detormine  the  time  of  describing 
a  given  portion  of  tiie  orbit ;  or,  conversely,  find  the  posi- 
tion of  the  particle  in  its  orbit  at  any  time.* 

♦  Sec  Talt  and  SiccWm  UynnmlcM  of  a  Particle,  p.  110;  alKo  PiUI'k  Mecli's. 


TUB  SECTIONAL    AREA. 


325 


(9) 


(joK.  4. — If  p  IS  the  periJendiciihir  from  the  origin  to 
the  tangent  wc  have  from  Calcuhis,  p.  170, 


which  in  (;3)  gives 
and  this  in  (0)  give 


xdy  —  y  dx  —  p  ds  ; 
h 


ds 
Jt 


P 


(10) 


.li^ 


(11) 


(ZA:  =  -2Pdr. 

r 

DiflFerentiating,  and  solving  for  P,  wo  have 

Zt2  dp 
f  dr' 

which  is  the  equnlion  of  the  orf)i/  Iwtwecn  the  radius  vector 
and  the  j)erpcnd tenia r  on  the  taiujcnt  at  any  point. 

182.  The  Sectorial  Area  Swept  over  by  the 
Radius  Vector  of  the  Particle  in  flny  time  is  Pro- 
portional to  the  Time.— Let  A  denote  this  area;  then  we 
have  from  Calculuo,  p.  ^(i-i, 

A  =  i ./'  ?•«  de 

=  I  ./•  h  dt,  by  (5)  of  Art.  181, 

if  A  and  t  be  botli  measured  from  the  commencement  of 
tiie  motion.  Therefore  the  areas  swept  over,  try  the  radius 
vector  in  different  times  are  proportional  tn  the  times,  and 
equal  areas  will  be  described  in  equal  times. 

foR.— If  /  =  1,   we  have  A  =  ^fi.     Hence  h  =  twice 
the  sectorial  area  described  in  one  unit  of  time. 

183.  The  Velocity  of  the  Particle  at  any  Point 
of  its  Otbit— Wc  have  Cur  tlie  velocity, 


3^0  VELOCITY  AT  AM'  l'Ol.\T  OF  THE   OKIilT. 

ds 


V  = 


dt 


=      by  (10)  of  Art.  181.         (I) 

Hence,  the  velocity  of  the  purticlv  al  each  jmint  of  its 
pat/i  is  inversely  proportional  to  the  perpendicular  from  the 
centre  on  the  tangent  at  that  point. 

Cor.  1. — We  liave,  by  Calculus,  \).  180, 


1 

1           1     rf/^ 

i^~ 

?-2  '^  r*  dm 

= 

"  -i^  aoi^  since 

which 

in 

(1) 

give? 

« 

V^  : 

//2 

=  -{'"  + a. 

(a) 


another  important  expression  for  the  velocity. 
Cob.  3.— From  (G)  of  Art.  181,  we  have 


(a) 


Let  V  be  the  velocity  at  the  point  of  projection,  at 
which  let  /•  =:  R,  and  since  P  is  some  fi'.nction  of  r,  let 
P  =  f{r),  tlien  integrating  (3)  we  get 


%=-,Jlf(r)dr, 


(•1) 


which  is  another  expression  for  the  velocity  ;  and  since  this 
is  a  function  only  of  the  corresponding  distances,  II  and  /•, 
it    follows   that  the   I'vlocity  at  any  point  of  the  orbit  is 


l:jii 


VELOCITY  AT  A.\Y  J'OI.\T  UF  THE   ORBIT. 


337 


(I) 


(a) 


(3) 


{^) 


iiulependeiU  of  //le  pal  ft  ikscribi'd,  and  depends  solely  on  the 
mnynilude  of  the  attraction,  the  distance  of  the  point  from 
llie  centre,  and  the  velocity  and  dixiance  of  projection. 

From  (4)  it  appears  tliat  tlie  velocity  is  the  same  at  all 
points  of  tlio  same  orbit  which  are  equally  distant  from  the 
centre;  if  r  =  /.',  the  velocity  =  T;  and  thus  if  the  orbit 
is  a  re-entering  curve,  the  particle  always,  in  its  successive 
revolutions,  passes  through  the  same  point  with  the  same 
velocity. 

If  the  velocity  vanishes  at  a  distance  a  from  the  centre 
(4)  becomes 

r-2  =  3[/i(fl)-A(r)]  (5) 

and  a  is  called  the  radius  of  the  circle  of  zero  velocity. 


(6) 


Cor.  3.— From  (3)  wo  have 

d{f)  =  -2Pdr; 

.•.    vdv  =  —  Pdr. 

Taking  the  logarithm  of  (1)  we  have 

log  V  =  log  h  —  log  p. 

Dilferentiating  we  get 

dv  _       dp 
~v  ~  ~  p' 

Dividing  (G)  by  (7),  wo  get 

''  =  ^P^lp  =  ^^-2Tp 
=  2P  x{  chord  of  curvature*  through  the  centre  ;    (8) 


an 


*  To  i)i()ve  that  \r<  ;)iU'-fouilli  the  elioid  of  curvature. 

•i  il/) 

Let  MI)  (Fig  HI),  be  llu^  liiiini'iit  to  tlie  orbit,  atid  0  the  centre  of  ciirvaliiri!  ;  let 

OD  =  i>.  CM  -  (),  tlie  radius  ofciirvotiire ;  and  tbo  angle  MEN  =  -A.    Then  MS.  the 


328 


VELOCITY  AT  AM'  POIST  OF  THE  ORBIT. 


and,  comparing  this  with  (C)  of  Art,  140,  it  appears  that 
tho  particle  at  any  point  haa  tlie  same  velocity  which  it 
would  liave  if  it  moved  from  rest  at  tliat  point  towards  the 
centre  of  force,  under  tiie  action  of  the  force  continuing 
eonstiint,  through  one-fourth  of  the  chord  of  the  circle  of 
curvature. 

Hence,  the  velocity  of  a.  p-rtiv(e  at  any  point  of  a  cei'tral 
orbit  is  the  mine  as  that  which  would  be  acquired  by  a 
particle  moving  freely  from  7 est  throuyh  one-fourth  of  the 
chord  of  curvature  at  that  point,  throuyh  the  centre,  under 
the  action  of  a  constant  force  whose  magnitude  is  eqiial  to 
thai  of  the  central  attraction  at  the  point. 

OoR.  4. — If  the  orhit  is  a  circle  having  the  centre  of  force 

part  of  the  radius  vector  CM,  which  ix  intercepted  by  the  circle  of  curvature  li 
calleii  the  chord  o/ntrratiire.    Its  value  is  (IcloriiiiiU'd  m  follows; 


We  have  (Fig.  81) 


li)  =  0  +  OMD 
=  e  4  Bln-l  - 


d*  =  rf»  + 


From  Calculus,  p.  180,  (10),  we  have 


rdp—pdr_ 


and 

Subtititiiting  C»)  in  (I)  we  get 


«t»  = 


pdr__ 
rVr'—p'' 

r'd»  rdr 


Vr"-p« 


d<i>~ 


dp 


But  CalcaluB,  p.  881,  we  have 
p 

r:  j\v  M3  (Fig.  81)  =  aMP  sin  OMD 


i'r'-p' 


IVP  dp 


-  'in  '    =  «/)    ~-  , 

/•  dp 


'y  (8) 


thr  chord  of  curvature  ;  therefore 

5~  =-  (/iiefou-.lh  the  chord  of  curvattire. 
%dp 


0) 

CD 
CO 

(«) 


THE   Oli-IT  bWDEH    VARIABLE  ATTRACTION.        oZ'd 

in  the  centre,  and   R,  T,  P,  are  respectively   the  radius, 
velocity  and  central  faro,  we  have 

F»  =  PR. 


0) 


(4) 


Cor.  5.— From  (5)  of  Art.  181,  we  have 

^  _  A 
dt  ~  »•»' 


{») 


The  first  member,  being  the  actual  velocity  of  a  point 
on  the  i-;uliiis  vector  at  the  unit's  distance  from  the  centre, 
is  the  angular  velocity  of  the  particle  (Art.  100).  Hence 
the  aiii/ular  velocity  of  a  particle  varies  inversely  as  the 
square  of  the  radius  vector. 

ScH.— A  point  in  a  central  orbit  at  which  the  radius 
vector  is  a  maximum  or  minimum  is  called  an  Apse ;  the 
radius  vector  at  an  apse  is  called  an  Apsidal  Distance  ;  and 
the  angle  between  two  consecutive  apsidal  distances  is  called 
an  Apsidal  Awjlf  of  the  orbit.     The  aaalytical  conditions 

(III 

for  an  ai)se  are,  of  course,  that  -j^  =  0,  and  that  the  first 

derivative  which  does  not  vanish  sliould  be  of  an  even 
order.  The  first  condition  ensures  Mat  the  radius  vector 
at  an  apse  is  perpendicular  to  the  tangc'it. 

184.  The  Orbit  when  the  Attraction  Varies  In- 
versely as  the  Square  of  the  Distance.— .1  particle  is 
projected  from  a  f/iven  point  in  a  r/iven  direciidn  vitli  a  i/iven 
relorify,  and  moves  under  the  action  of  a  central  attraction 
varying  inversely  as  the. square  of  the  distance;  to  determine 
the  orbit. 

Let  tbecciilrc  of  forc"  !)e  the  origin  ;  I' =  the  velocity 
of  projection  ;  A'  =  'ho  distiMU'C  of  the  point  of  i)rojecti()n 
from  tlio  origin;  (i  =--  the  angle  between  R  and  the  line  of 


330        TUB  ORBIT  UNDER    VARIABLE  ATTRACTIOK. 

projection ;   and  let  fi  =  the  absolute   force  and    /  --  0 
when  the  particle  is  projected.     Then  since  the  velocity  = 

(Art.  183),  and  at  the  point  of  projection  _p  =  ^  sin  (i, 

we  have 

V  =  ,-i-^  ;  h=VR  sin  /3.  (1) 

R  sin  p 

As  the  force  varies  inversely  as  the  square  of  the  distance, 
we  have 

P  =  "^  =  (iifi,  (since  r  =  -)•  (2) 


wh'ch  in  (9)  of  Art  181  gives 
tPu 


5^  +  "  =  W 


|3) 


Multiplying  by  2du  and  integrating,  we  get 


%^u^  =  2lu^c; 


F2 


1  1  du^  r  - 

when  /  =  0,  M  =  -  =  -^,  and  ^^  +  «*  =  -p-,  (Art.  183, 
Cor.  1 ) ;  therefore 

'^  -  Iv'  ~  WR  ~        li'R 


Substituting  this  value  for  c  we  get 


V^R  -  %\i      2uu 


+    W'   =   — T-o-f,  -^   + 


A^ii; 


Therefore  (Art.  183,  Cor.  1)  we  have 

(velocity)-^  =  ^''  +  "''^C-i) 


(4) 


(5) 


m 


THE  ORBIT   LWDEU    VAlilAllLE  ATTRACTION.        331 

which  shows  that  (he  velocity  is  the  greatest  when  r  is  the 
least,  anil  the  least  when  r  is  the  greatest. 
Changing  the  form  of  (4)  we  have 


V^R  -  2ti      /i 


h^R         '  h* 
To  express  this  in  a  simpler  form,  let 


-(£-«)• 


(6) 


h' 


-  =  b,  and 


V^R 


2/t 


h^R 

dm 


-I-  ^  =  c* ;  and  (6)  becomes 


¥ 


=  c2 -(«-&)»; 


—  du 


[C2  _  (m  _  J)2]* 


=  de, 


the  negative  sign  of  the  radical  being  taken.     Integrating 

we  have, 

_,u  —  b  , 

cos  * =0  —  Cf 


(4) 


(5) 


where  c'  is  an  arbitrary  constant; 

.  • .    u  =  b  +  c  cos  {6  —  c'). 


(7) 


Replacing  in  (T)  the  values  of  b  and  c.  and  the  value  of  h, 
from  (1),  and  dividing  both  terms  of  the  second  member  by 
ft,  we  have  for  the  equation  of  the  path. 


1  + 


u  = 


LF' 


{V^R-2n)  RVUm^fi  +  1 


cos(fl— c') 


/r^  v^  sin'^  a 


(«) 


which  is  the  eciuation  of  a  conic  section,  the  pole  being  at 
the  focus,  and  the  angle  {(>  —  c')  being  measured  from  the 


m 


3.'{2        TBE   OliBlT   rXDEIi    VAIilMil.K  ATTIiACTIOX. 

shorter  length  of  the  uxis  major.  For  if  e  is  the  eccentricity 
of  ii  conic  section,  r  the  focal  mdius  vector,  and  </>  the 
angle  between  r  and  tliat  point  of  a  conic  section  which  is 
nearest  the  focus,  we  have. 


1  _      _  ^  +  c  cos  ^ 
r  ~      ~      1  ~  e^     ' 


(9) 


Comparing  (8)  and  (9),  we  see  that 


ca  =  -^  (  F2/.'  -  -ifi)  RV^  8in2 13  +  1; 


(10) 


(f)z=d  —  c'. 


(11) 


Now  the  conic  section  is  an  ellipse,  parabola,  or  hyper- 
bola, according  as  e  is  less  than.  ci|iuil  to,  or  greater  tlian 
unity;  and  from  (10)  c  is  less  than,  cfjual  to,  or  greater 
than,  unity  according  as  V^K  —  'ifi  is  negative,  zero,  or 
positive ;  therefore  we  see  that  if 


%H 


F*  <  -^,  e  <  1,  and  the  orbit  i§  an  ellipse,         (18) 


2u 
V^  =  -py  e  =  1,  and  the  orbit  is  a  parabola,        (13) 

F'  >  -^,  c  >  1,  and  the  orbit  is  a  hyperbola.      (14) 


CoK.  1. — By  (1)  of  Art.  173,  wq  see  that  the  square  of 
the  velocity  of  a  pa  icle  falling  from  infinity  to  a  distance 
R  from  the  centre  of  force,   for  the  law  of  attraction  we 


arc  considering,  is 


// 


Hence   the  above  conditions   tmuv 


be  expressed    more  concisely    by   saying   that    f/ir   nrhif, 
described  about  Ihi^  centre  of  force,   will  be  an  ellip.ye,  a 


ftX. 


THE  ORUIT  Aa\  ellipse. 


3:53 


cx'iitricity 

id   <f»   till' 

which  is 


(9) 


(10) 

(11) 

or  hyper- 
3iitcr  than 
)r  great  or 
,  zero,  or 

(13) 
.,        (13) 

hi.      (14) 

square  nf 
I  distance 
action  we 

ons    niay 

'he   iir/iif. 
cifip.sr,  a 


parabola,  or  a  hypcrbohi,  tiaonliiKj  as  thv  vflority  is  lean 
ifian,  equal  to,  or  (jrvulvr  than,  the  velocity  from  injinity. 

The  speriis  of  conic  section,  tliereforc,  docs  not  depend 
oil  the  po.-ilion  of  tlic  line  in  wliich  tlic  particle  is  pro- 
jected, but  on  the  velocity  of  projection  in  reference  to  the 
distance  oi  the  point  of  projection  from  the  centre  of 
force. 

Cor.  2. — From  (11),  we  see- that  0  —  c'  is  the  angle 
between  tlie  focal  radius  vector, 
r,  and  that  i)art  of  the  principal 
axis  which  is  between  tiie  focus 
and  the  jioint  of  the  orbit  wliicli 
is  nearest  to  the  focus ;  /.  e.,  it 
is  the  angle  PFA  (Fig.  82) ;  and 
therefore  if  tiie  principal  axis  is  the  initial  line  c  =z  0. 

185.  Suppose  the  Orbit  to  be  an  Ellipse. — Here 
r*  <  "^' ;  so  that  from  (10)  we  have 

e«  =  1  -  yj  {2ii  -  r^B)  R  V^  sin*  j3.  (1) 

Now  the  ecpiation  of  an  ellipse,  where  /•  is  the  focal 
radius  vector,  B  t!ie  angle  between  /■  and  tiic  shorter  seg- 
ment of  the  major  axis,  2«  the  major  axis,  e  the  eccen- 
tricity, is 

_  a  (1  -  e")  , 

^  ~  \  +  ecosfl' 


u  = 


^  + 


e  cos  0 


fl(l  _e2)   '    a  {I  -e«)' 
comparing  (2)  with  (8)  of  Art.  184,  we  have 

_!„    _        /^ . 


(2) 


m 


334 


y/y/i'   OUIilT   AN  ELLIPSE. 


substituting  for  J  —  e^  its  value  from  (1),  and  solving  for 
a,  we  have 

_        \iR  .  , 

which  tyiows  that  the  major  axis  is  independent  uf  tfie  direc- 
tion of  project  ioti.  ' 

We  may  explain  the  several  q.iantities  which  we  have 
used,  by  Fig.  S'i. 

B  is  the  point  of  projection;  FB  =  R;  DB  is  the  line 
along  Avhich  (ho  particle  is  projected  with  the  velocity  V; 
KBD  =  f3,  the  angle  of  |)rojecti()n ;  FP  =  r;  PFA  =  6, 
FD  =:  R  sin  li ;  if  (3  =  90",  the  particle  is  projected  from 
an  apse,  i.  e.,  from  A  or  A'. 

CoK.  1. — To  determine  the  apsidal  distances,  FA  and 
FA',  we  must  put  '^."  =  0,  (Art.  183,  Sch.),  and  (4)  of 
Art.  184  give  us  the  quadratic  equation 


u* 


■ifi 


ya 


h 


'\i "  +  Wr  ~  li'i  -  ^' 


14) 


the  two  roots  '<f  which  arc  the  recijirocnls  of  the  ttvo  apsidal 
distances,  a  {1  —  r)  and  ^^  (1  -4-  c). 

Coiv.  2.— Since  the  coetticient  of  the  second  term  of  (4) 
is  the  sum  of  the  roots  witii  their  signs  changed,  wo  have 


1 

-)  + 

1 

0(1- 

«  (1  + 

e)- 

•      • 

ail- 

-^)  = 

2^. 


(6) 


which  (/ires  /he  l(ih(s  rectum  of  the  orbit. 


rtMiiAa 


KEPLEI{  'S  LA  WS.  335 

Cor.  3.— From  Art.  182  we  have,  calling  T  the  time, 

2A 


T  = 


V 


where  A  is  the  area  swept  over  by  the  radius  vector  in  tiio 
time  T.     Therefore  for  the  lime  of  describing  an  ellipse, 

we  have 

rp  _'i  area  of  ellipse 


(5) 


_  %-rxd>-  Vl  -  «» 
'^a\i.  (1  —  e^j 


,  from  (5), 


which  is  the  time  occupied  by  the  particle  vn  passing  frmt 
any  point  of  tlie  ellipse  around  to  the  same  point  again* 

186.  Kepler's  Laws.— By  lal)o)ious  calculation  from 
an  immense  series  of  observations  of  the  planets,  and  of 
Mars  in  particular,  Kepler  i'nunci.i'.ed  the  following  as  the 
laws  of  the  planetary  motions  about  the  Sun. 

T.  The  orbits  of  the  phinets  are  ellipses,  of  which 
the  Sun,  occupies  a,  focus. 

IT.  Tlie  radius  rector  of  each  planet  describes 
equal  areas  iu  equal  times. 

in.  The  squares  of  the  periodic  times  of  the 
planets  are  as  the  cubes  of  the  major  a.ves  of  their 
orbits. 

187.  To  Determine  the  Nature  of  the  Force  which 
Acts    upon   the    Planetary  System. -(1)    Froin   the 

♦  Called  feriixilc  Time. 


336 


J'LAAET.l  /.'  }■  S  rsTEM. 


second  of  these  laws  it  follows  that  the  planets  are  rctuinod 
in  their  orbits  hy  an  uttraetion  tending  to  the  Hun. 

Let  {x,  y)  be  tiie  jjosition  of  a  planet  at  the  time  t 
referred  to  two  eo-ordinate  axes  drawn  through  the  Sun  in 
tiie  plane  of  motion  of  the  planet;  X,  Y,  the  component 
accelerations  due  to  tiie  attraction  acting  on  it,  resolved 
parallel  to  the  axes;  then  the  equaticnis  of  motions  are 


iPy  _ 


r; 


dp 


(I    30  -wr-  TT 


0) 


But,  by  Kepler's  second  law,  if  A  bo  the  area  described 
(lA 
dt 


by  the  radius  vector,  -^  is  constant, 


rfA 


r^de 


^    dt 
=  *(^!/f  "•'^fl)  =  a  constant. 


Diflfercntiating,  we  have 


'dp 


iPx 


^J7^-!/dP=^- 


xV-  !/X  =  0,  fVom(l), 


X 

r 


X 


which  shows  that  the  axial  components  of  the  acceleration, 
due  lo  the  atiraedon  acting  on  tiie  j)lanet,  are  proportional 
Id  the  cD-ordiiKitcs  of  Die  |)hinet;  and  therefore.  I)y  the 
parallelogram  of  forces  (Art.  ;5<M,  the  resultant  of  A' and  )' 
passes  Ihrougli  the  origin. 


PLAXETAUr  SYSTEM. 


337 


ire  retuinotl 
un. 

the  time  i 
the  Sun  in 

component 
it,  resolved 
ous  are 


(1) 

a  described 


icceleriition, 
liropnrtioniil 
lore.  I)y  the 
of  A'liiul  )' 


Hence  l/ie  forces  actinj  on  the  phmels  all  pass  Ihrotujh 
the  Sun's  centre. 

(2)  From  tlic  first  of  these  laws  it  follows  that  the 
central  attraction  varies  inversely  as  the  square  of  the 
distance. 

The  polar  equation  of  an  ellipse,  referred  to  its  focus,  is 

_    «(1  -  e'l^ 
'"  —  1  4-  e  cos  0' 

1  +  fc'  cos  0 
(Pti    ,       _  1 . 


or 


Hence 


u  = 


and  t'lereforo,  if  /'  is  the  attraction  to  the  focus,  we  have 
[Art.  181,  (9)], 

h^        1 


~  a{l-e^)  »•* 

Hence,  if  the  orbit  be  an  ellipse,  described  about  n  centre 
of  attract  ion  at  the  focus,  tlie  law  of  intensity  is  tliat  of  the 
inverse  square  of  the  distance. 

{'])  From  the  third  law  it  follows  that  the  attraction  of 
the  Sun  (.supposed  fixed)  wliich  acts  on  a  unit  of  mass  of 
each  of  the  planets,  is  the  same  for  each  planet  at  the  same 
distance. 

By  Art.  185,  Cor.  3,  we  have 


4tt3 
T»  =  ---  a^ 
1^ 


U 


338 


EXAMPLES. 


But  by  the  third  law,  7^  oc  u%  iind  therefore  w  must  be 
constant;  i.  e.,  the  strength  of  attraction  of  tlie  Sun  must 
be  the  same  for  all  the  planets.  Hence,  not  only  is  the  law 
of  force  the  same  for  all  the  plaufcts,  but  the  ahsohde  force 
is  tlie  same. 

Tl)i.s  very  brief  discussion  of  central  forces  is  all  that  we 
have  space  for.  To  pursue  these  enquiries  further  would 
com])el  us  to  omit  matters  that  are  more  especially  entitled 
to  a  place  in  tliis  book.  The  student  who  wisiies  to  pursue 
the  study  further  is  referred  t^o  Tait  and  Steele's  Dynamics 
of  a  Particle,  or  Price's  Anal.  Mech's,  Vol.  I,  or  to  any 
work  on  Mathematical  Astronomy.  We  shall  conclude 
witii  the  following  examples. 


EXAMPI.  ES, 

1..A  particle  describes  an  ellipse  under  an  attraction 
always  directed  to  the  centre  ;  it  is  required  to  find  the  law 
of  the  attraction,  the  velocity  at  any  point  of  the  orbit,  and 
the  periodic  time. 

(1)  The  polar  equation  of  the  ellipse,  the  pole  at  the 
centre,  is 

cos8(?  .   sin^e 


u^  = 


a" 


+ 


But  [Art.  181,  (!))]  we  have 


(1) 
(2) 

(•'0 


EXAMPLES. 


339 


w  must  be 
3  Sun  must 
y  is  the  law 
solute  force 

all  that  we 
ther  would 
lly  entitled 
'S  to  pursue 
Dynaniies 
or  to  any 
1  conclude 


attraction 
ind  the  law 
J  orbit,  and 


pole  at  the 
(1) 
(2) 

0.  (3) 


by  (3), 

(cos'  d  -  sin^  e)],  by  (2), 
by  factoring, 


(4) 


ami  th(>rcfore  the  attraction  varies  directly  as  the  distance. 
If  ft  =  the  absolute  force  wc  have,  by  (4), 

/t2  -  ft  (W.  (5) 

(2)  If  V  =  the  velocity,  wc  have,  by  Art.  183, 

,ii  =  -!  =  -iS^(Ana].  Geom.,  p.  133) 

=  ltb'%  by  (5), 
where  b'  is  the  semi-diameter  conjugate  to  r. 
,  • .    V  =  0'  V/*. 

(3)  U  T  =  the  periodic  time,  we  have,  by  Art.  182, 

and  iu'iice  tlie  periodic  time  is  independent  of  the  magni- 
tude of  the  ellipse,  iind  depends  only  on  the  absolute 
central  attraction.    (Sec  Tait  and  Steele's  Dynamics  of  a 


^IM 


340 


EXAMPLES. 


Particle,    p.    144,    also    Price's    Anal.    Mech's,    Vol.    I, 
p.  516.) 

2.  A  particle  describes  an  ellipse  under  an  attraction 
always  directed  m  one  of  the  loci  ;  it.  is  retiiiired  to  find  tiie 
law  of  attraction,  the  velocity,  and  the  periodic  time. 


(1)  Here  we  have 

1  +  '^  cos  0 


u  = 


r,'\       ' 


and 


(Pii 


(lu  _ 
■    dd  ~  ' 

—  e  cos  0 


e  sin  0 


(1) 


which  in  ('.))  of  Art.  181  gives 


F  = 


a(i  -(»)       a  n  —  £•«)    r»' 


(2? 


hence  the  attraction  varies  invcr.vly  as  the  stpiare  of  the 
distance.     If /i  =  tiie  absolute  force,  we  have  by  (2) 

//2  =  lia  (1  -  e*).  (a) 

(2)  By  Art.  183,  Cor.  1,  we  have 

1  „      (lit^        2ai<.  —  1     ,     /^v  ,.\ 

^-«*  +  ;?ff. -,,^7r3.-,i)'V(i);         (4) 

A^       fi^iau  —  1)    ,        .       ,     . 
•'•  p^  ^  It         '   ^^  ^^  ""    ^^'  ^^ 

(3)  If  r  =  the  iHTJodic  time  wo  have  (Art.  182) 

_  'i^^  (J.  -  ^t 
"         h' 


m  attraction 
d  to  find  tlie 
c  time. 


--  •        (1) 


(8? 

iliiare  of  tlie 

(3) 


);         (4) 

(4).  (5) 

182) 


«», 


(G) 


I's,    Vol.    I,       I 


,1 


kXAMI'I.KS. 


d4i 


and  hence  the  periodic  time  vir.-ic°  as  the  sf|uare  root  of 
tlie  cube  'f  the  major  axis. 

3.  Find  the  attraction  l»y  whicli  a  particle  may  describe 
a  circle,  and  also  the  velocity,  and  the  periodic  time,  (1) 
when  the  centre  of  attraction  is  in  the  centre  of  the  ciiclc. 
and  (3)  when  the  centre  of  attraction  is  in  the  circiiiii- 
ference. 

(1)  Let  a  =  the  radins;  then  the  polar  equation,  the 
pole  at  the  centre,  is 


r  =  «; 


1     (ht.      fPu  __  - 


^  =  ^^'K«+S)  =  ^ 


r.3" 


Also 


,.«=  -.,     and     y=   -ir- 


(1) 
(2) 


From  (1)  and  (3)  we  have 


P  =  !?, 

a 

and  hence  the  central  attraction  is  etpial  to  the  square  of 
the  velocity  divided  )>y  the  radius  of  the  circle.* 

(2)  The  equation,  is 

r  =  2a  cos  d  ;     .  • .    ;iau  =  sec  fl. 


and 


«  +  TS  =  8a''?*' ; 


.-.    P  =  8fl%«7t»  =  ^' ; 


and   hence   the  attractioi\    varies   inversely    as    the    fifth 

*  Called  \\v\  <'ei)tHptgtil  Fin\    Sec  Art  108. 


342 


EXAMPLES. 


power  of  the  distance  ;   and  if  /'  =  the  absohite  toro-,  w« 
iiiive  n  =  SuVi^; 


A« 


and     ?*  — 


Zi" 


If  7'  ==:  the  periodic  time,  we  have 


T  ^  -^1™'.     (See  Price's  Anal.  Mech.,  Vol.  I.,  p.  618.) 

4.  Find  the  attraction  by  which  a  particle  may  describe 
the  lemniscatc  of  Bcrnouilli  and  aKso  the  velocity,  j,nd  tiie 
time  of  describing  one  loop,  the  centre  of  attraction  being 
in  the  centre  of  the  lemniscate,  and  the  equation  being 
'/•2  =  a«  cos  20. 

„....  =  5'^;  .^  =  ,ii;r=©V. 

5.  Find  tlie  attraction  by  which  a  particle  may  describe 
the  cardioid  and  also  the  velocity,  and  the  periodic  time, 
the  equation  being  r  z=  a  {I  +  cos  6). 

Ans,  P  =  -^'^  ^'  =  ^>    '  =  \-y)''- 

(].   Find  the  attraction  by  which  a  particle  may  describe 

a  luirabola,  and  also  the  velocity,  the  centre  of  attrpction 

.       ,    -  2a 

being  at  tl.e  focus,  and  the  equation  bemg  r  =  ■        ^-y 


Ans.   P  - 


2ar* 


f2  = 


Compare  (13)  of  Art.  184. 


T.  Find  the  attraction  by  which  a  particle  may  describe 

■A  hyperbola,  and  the  velocity,  the  centre  of  attraction  l)eing 

.      ,   .  a(«2-l) 

at  the  focuc,  and  the  efpuition  bemg  r  =  i-JT^-^fQ 

h^        1       „       fi  (2fl»  +  1) 

Am.   P  =      ,,    — sr  ;j;  '^  = • 

a  (1  —  e*)  r^  a 


te  for'  p.  w« 


p.  6l8.i 

nay  describe 
iity,  j,nd  the 
letion  being 
aation  being 

=  ©*- 

nay  describe 
iriodic  time, 


h 


may  describe 
)f  attrpction 

_  2a ^ 

~  1  +  cos  O' 

f  Art.  184. 

may  doscribt) 
ruction  l)eing 

'■-JL 

e  cos  6* 
{2aii  +  1) 


exampi.es. 


34:} 


8.  If  the  centre  of  utt'-.  .aion  is  at   tlic  centre  of   the 
liyperbohi,  find  tlio  attraction,  and  velocity,   the  c'luation 


beins' 


cos'^  6       sin^  0 


b^ 


=    H' 


ir- 


Ans.    "=.--j^r  = 


fir;  v^  —  fi  (r^  —  a^  +  //*). 


9.  Find  the  attraction  to  the  pole  under  which  a  particle 

will  describe  (1)  the  curve  whose  equation  is  /•  =  ia  cos  nd, 

2a 

and  (2)  the  curve  whose  equation  is  r  = „• 

^  '  '  1  —  e  cos  nO 


¥n- 


•  .e! 


Ans.  (1)  /'  =       j^—  +  ' ^-~   ;     {2)F  -  ^^^ 

> '  —  •     That  is,  the  attraction  in  the  first  curv 

partly  as  the  inverse  fiftii  [.  )wer,  and  partly  as  th<!  .iveioO 
cube,  of  the  distance  :  and  in  the  second  it  varies  ;  :,  'y  • . ' 
the  inverse  square,  and  partly  as  the  inverse  cube,  i  ihe 
distance. 

10.  A  planet  revolved  round  the  sun  in  an  orbit  with  a 
major  axis  four  times  that  of  tiie  earth's  orbit ;  determine 
the  periodic  time  of  tiie  ])lanet.  Ans.  8  years. 

11.  If  a  satellite  revolved  round  the  earth  close  to  its 
surface,  determine  the  periodic  time  of  the  satellite. 

of  tlie  moon's  period. 


A71S. 


(60)« 


12.  A  body  describes  an  ellipse  under  the  action  of  a 
force  in  a  focus  :  compare  the  velocity  when  it  is  nearest 
the  focus  with  its  velocity  when  it  is  furthest  from  the 
focus. 

Ans.  As  1  +  e  :  1  —  ;',  where  n  is  the  eccentricity. 

13.  A  Vdy  describes  an  ellipse  under  the  .iction  of  a 
force  to  the  focus  ,S' ;  if  //  l)e  the  other  foois  show  that  the 


^ 


344 


KXAMPLKS. 


velocity  at  any  point  /'may  he  resolved  into  two  velocities, 
resi)e"tively  at  riglil  angles  to  A'/' and  ///',  and  each  vary- 
ing ilS  III*. 

14.  A  body  deseriltes  an  ellipse  under  tlie  aetion  of  a 
force  in  t lie  eentre:  if  the  greatest  velocity  is  three  times 
the  least,  find  the  eccentricity  of  the  ellipse.  Am.  |  v-'. 

• 

15.  A  body  d<  scribes  an  ellipse  under  the  action  of  a 
force  in  the  centre  :  if  tlie  major  axis  is  20  feet  and  the 
greatest  velocity  2U  feet  per  second,  find  the  periodic  time. 

Ans.  n  seconds. 

16.  Find  the  at  tract  i<m  to  the  pole  under  which  a  par- 
ticle mav  descrihi'  tin  eipiiangular  spiral.  ,         ,,      1^ 

17.  If  P  =-  (oj-^  —  Hr),  and  a  particle  be  projected 
from  an  apse  at  a  distance  c  with  tlie  velocity  from  infinity  ; 
prove  that  the  equation  of  the  orbit  is 

18.  If  P  =  2u  (     —  "X  'i'>J  '''^'  particle  be  projected 

from  an  ap^ie  at  a  distance  n  with  velocity     -  ,  prove  tiiat 
it  will  be  at  a  distance  r  after  a  time 

1     /  , ,      r  +  Vr«  —  ffl*  ,   ^ 


a 


Vr^  -  «')• 


velocities, 
fiicli  vary- 

actioii  of  a 
tliroc  tiiiu's 
I  V-i. 

action  of  a 
et  and  the 
iodic  time, 
seconds. 

hicli  a  par- 
1 


*.    Pr. 


r3 


le  jirojected 
Dm  infinity ; 


)e  projected 
,  prove  that 


a«). 


CHAPTER    III. 

CONSTRAINED     MOTION. 

188.  Definitions. — A  particle  is  cnnslrnined  in  its  mo- 
tion when  it  is  compelled  to  move  ahmg  a  given  fixed  curve 
or  surface.  Thus  far  the  sul)jeets  of  motion  have  been 
particles  not  constrained  by  any  geometric  conditions,  but 
tree  to  move  in  such  jiaths  as  arc  due  to  the  action  of  the 
impressed  forces.  We  come  now  to  tiie  case  of  the  motion 
of  a  particle  which  is  constrained ;  that  is,  in  which  the 
motion  is  subject,  not  only  to  given  forces,  but  to  undeter- 
mined reactions.  Such  cases  occur  when  the  particle  is  in 
u  small  tube,  eitiier  smooth  or  rough,  the  bore  of  which  is 
supposed  to  be  of  the  same  size  as  the  particle ;  or  when  a 
small  ring  slides  on  a  curved  wire,  with  or  without  friction  ; 
or  when  a  particle  is  fastened  to  a  string,  or  mo\es  on  a 
given  surface.  If  we  substitute  for  the  curve  or  surface  a 
force  whose  intensity  and  direction  are  exactly  equal  to 
those  of  the  reaction  of  the  curve,  the  particle  will  describe 
tiie  same  path  as  l)efore,  and  we  may  treat  the  problem  as 
if  the  particle  were  free  to  move  under  the  action  of  this 
system  of  forces,  atid  therefore  api)ly  to  it  the  general  equa- 
tions of  motion  of  a  free  particle. 

189.  Kinetic  Energy  or  Vis  Viva  (Living  Force), 
and  Work. — A  particle,  in  const  mined  to  move  on  a  given 
smoutli  plane  ritrre,  under  given  forces  in  the  plane  of  the 
curve,  to  determine  the  motion. 

Let  APC  be  the  ciTtve  along  which  the  particle  is  com- 
pelled to  niove  when  acted  upon  by  any  given  forces.  Let 
0.C  and  0^  be  the  rectangula.'  axes   in    the  plane  of  tiic 


346 


KINKTlC  KSEUdV. 


curve,  the  axis  y  positive  up- 
wards, aud  (X,  y)  tlic  place  of 
tiie  particle,  /',  at  tlic  time  / ; 
lot  X,  Y,  parallel  respectively  to 
the  axes  of  .r  ami  //.  he  the  axial 
components  of  the  forces,  the 
mass  of  the  jiarticle  being  m  ; 
let  li  be  the  pressure  between 
the   curve   and    particle,    which 

acts  in  the  normal  to  the  curve,  since  it  is  smooth.     Then 
the  equations  of  motion  are 


Fig.83 


di^  as 


(1) 


(Py  dx 


dP 


ds 


(2) 


Multiplying  (1)  and  (2)  respectively  hy  dx  and  Jy,  and 
adding,  we  have 

dx  d^x  +  du  d^ti        „ ,     ,    „ , 


Integrating  between  the  limits  t  and  t^,  and  calling  v^  tho 
mitial  velocity,  we  have 


m 


_^V=   f{Xdx+Ydy) 


(3) 


The  term  -^  v^  is  called  the  vis  viva*,  or  Kinetic  Energy 

of  the  mass  m ;  that  is,  vis  viva  or  kinetic  energy  is  a 
quantity  which  varies  as  the  product  of  the  mass  of  the 
particle  aud  the  square  of  its  velocity.  There  is  particular 
advantage  in  defining  vis  viva,  or  kinetic  energy,  as  hrdf 


See  Thomson  and  Tail's  Nat  Phil.,  p.  288. 


th.    Then 

(1) 

(2) 

nd  dy,  and 


ing  Vq  tlifl 


(3) 


ic  Energy 

iicrgy  is  a 

uss  of  the 

particular 

^y,  as  hnlf 


RINETIf  EXKRIIY. 


341 


tho  product  of  the  nia,*s  and  tiio  sijuaro  of  its  velocity.* 
Tiic  lirst  nieml)cr,  therefore,  of  (3)  is  the  vis  viva  or  kinefi( 
energy  of  »(  ii((|uirfd  in  its  nn.tion  from  (,'•„.  //,,)  to  (.r,  //) 
under  tlie  action  of  the  given  fonvs. 

Tlie  terms  Xdx  and  Vdy  arc  the  products  of  the  axial 
components  of  the  forces  by  the  axial  displacements  of  the 
mass  in  the  time  dt,  and  are  therefore,  the  elements  of  work 
done  hy  the  acceleniting  forces  X  and  V  in  the  time  <//, 
according  to  tiie  delinition  of  work  given  in  Art.  101,  Rem.; 
so  that  the  second  member  of  (3)  expresses  the  work  done 
Ity  these  forces  through  the  spaces  over  which  they  moved 
the  mass  in  tlie  time  between  /„  and  /.  This  equation  is 
QnWinX  the  equation  of  ki net ir  eneriiy  and  of  work ;  it  shows 
that  the  work  done  by  a  force  exerting  action  through  a 
given  distance,  is  !(|ual  to  the  increase  of  ki  ■  tic  energy 
which  has  accrued  to  the  mass  in  its  motion  through  that 
distance. 

If  in  the  motion,  kinetic  energy  is  lost,  negative  work  is 
done  by  the  force  ;  ('.  c,  the  work  is  stored  up  as  potential 
work  in  the  mass  on  which  the  force  has  acted.  Thus,  if 
work  is  s|)ent  on  winding  up  a  watch,  that  work  is  stored 
in  the  coiled  si)ring,  and  is  thus  i)otential  and  ready  to  be 
restored  under  adapted  circumstances.  Also,  if  a  weight  is 
raised  through  a  vertical  distance,  work  is  spent  in  raising 
it.  and  that  work  may  be  recovered  by  lowering  the  weight 
throuiih  the  same  vertical  distance. 

This  theorem,  in  its  most  general  form,  is  the  modern 
jirinciple  of  conservation  of  energy  :  and  is  made  the  funda- 
mental theorem  of  abstract  dynamics  as  applied  to  natural 

philosophy. 

In  this  case  we  have  an  \n»tanci'  of  spare-integrals,  which, 
as  we  have  seen,  gives  us  kinetic  energy  and  work  ;  the 
soli  >n  of  problems  of  kinetic  energy  and  work  will  be 
explai.ied  in  Chap.  V. 

*  Some  writers  defliie  vis  viva  a-*  the  whole  product  of  the  ina^o  and  the  equaw 
of  the  vclocily.    See  RouthV  Rigid  I)yiiam|i».  p.  259. 


* 


•M9,         RBAOTIOK  OF  TUE   COSSTiiAlMXa    CUttVE. 

Now  if  .rand  J'  arc  functions  of  the  co-ordinates  x  and 
1/  the  second  momhe-  of  (;5)  can  he  integrated  ;  let  it  he  the 
differential  of  some  fiinetion  of  .r  and  //,  as  (/>  {x,  y).  Inte- 
grating (3)  on  this  hypothesis,  and  sui)posing  r  and  v^  to 
be  the  velocities  of  the  particle  at  the  points  {x,  ij)  and 
(a:,,  ?/u)  corresponding  to  /  and  Z^,  we  luive 


m 


(t;2  _  j,„2)  =  0  (.r,  y)~-4>  (Xo,  v/o) 


(4) 


which  siiows  that  the  kinetic  energy  gained  by  the  particle 
constrained  to  move,  nnder  the  forces  J:',  F,  along 
any  path  whatever,  from  the  point  {x^,  y^)  to  the  point 
{x,  y),  is  entirely  independent  of  the  patii  pursued,  and 
depends  only  upon  tiie  co-ordinates  of  the  points  left  and 
arrived  at;  the  reaction  R  does  not  appear,  which  is  clearly 
as  it  should  be.  since  it  does  no  work,  because  it  acts  in  a 
line  perpendicular  to  the  tlirection  of  motion. 

190.  To  Find  the  Reaction  of  the  Constraining 
Cxirve. — For  convenience,  the  mass  of  the  particle  may  bo 
taken  as  unity.     Multiplying  (1)  aiul  (2)  of  Art.  189  by 

f-  and  'v.  subtracting   tl>e   former   from   the   latter,  and 
m  (Is 

solving  for  R,  we  have, 
iPif  dx 


........    ^dh^ly        ydy 

^^  ~  ,'lPch  '^      ds 


lis 


-  H- 
P 


X'll  -  F^^,  by  (:{)  of  Art,  1G3    (1) 


in  which  p  is  the  radius  of  curvature  at  the  point  P.  The 
last  two  terms  of  (1)  are  the  normal  components  of  the 
impressed  forces;  and  Iherciore.  if  the  ])artiele  wi're  at  rest, 
they  would  denote  the  whole  pressure  on  tiie  curve;  but 


iii:ites  X  and 
'vA  it  bo  th(? 
I-,  ?/).  Intc- 
V  iiml  t'j  to 
{X,  y)  iiiid 


BEACTIOS  OF  TirK   t'OySTRAINING    CURVE. 


349 


(4) 


the  particle 
,  F,  along 
3  the  point 
ursueil,  and 
ints  left  and 
cli  is  clearly 
it  acts  in  a 


>nstraining 

tick'  may  bo 
Art.  189  by 

latter,  and 


It 

'h 

^rt.  102    (1) 

int  r.    The 

lents  of  the 

were  at  rest. 

curve ;  but 


the  particle  being  in  motion,  there  is  an  additional  pressure 

V- 
on  the  curve  expressed  by  -  • 

In  the  above  reasoning  we  have  considered  the  particle  to 
be  on  the  concare  side  of  the  curve,  and  tiie  resultant  of  A' 
and  F  to  act  towards  the  convex  side  along  some  line  as  PI 
so  as  to  produce  pressure  against  the  curve.  If  on  the 
contrary,  tliis  resultant  iK'ts  towards  the  concave  side,  along 
J'F'  for  example,  then,  whether  tl.c-  pa>-ticle  be  on  the 
concave  or  conve.v  side,  the  pressure  agai".:;i   thf  curve  wid 

.,2 

be  the  difference  between      and  the  normal  resultant  of  X 

P 
and  Y. 

191.  To  Find  the  Point  where  the  Particle  Vill 
Leave  the  Constraining  Curve. -It  is  evident  tint  at 
that  point.  A'  —  0,  as  there  will  be  no  pressure  against  the 
curve.     Therefore  (1)  of  Art.  190  becomes 

p  (Is      ^    ds 

=  F'  COS  F'PB 

if  /"  bo  the  resultant  of  A^and  Y. 

.-.     i^  =  F'p  cos  F' PR 

=  )iF'- 1  chord  of  curvature  in  the  direction  PF'. 

Comparing  tliis  with  (6)  of  Art.  140,  we  see  that  fhe 
pailich'  icill  h'uvt'  tin:  curve  at  the  point  where  its  velocity  is 
such  «.s  woidd  he  produced  by  the  result ivit  force  then  actinfj 
on  it,  ifrontinved  ronsfun/  duviny  if.^  full  from  rest  throuyh 
u  s/ntc'e  et/i((d  to  {  of  the  chord  (f  vurvalure  parallel  to  that 
rrsultunt.  (See  Tait  and  Steele's  Dyuumics  of  a  Particle, 
p.  170.) 


350  CCiNSTUAfXED  MOTION. 

192.  Constrained  Motion  Under  the  Action  of 
Gravity. — When  gravity  is  the  only  force  acting  on  tlio 
particle,  the  formulae  are  simplified.  'J'aking  tiie  axis  of  y 
vertical  and  positive  downwards,  the  forces  become 

X=  0,    and     Y  =  +</; 

and  for  the  velocity  we  have,  by  (3)  of  Art.  189, 

\^^-\i\'  =  <j{y~y^)  (1) 

where  ?/,  is  the  initial  space  corresponding  to  the  time  t^. 
For  the  pressure  on  tiie  curve  we  have,  by  (1)  of  Art.  190, 

If  the  origin  be  where  the  motion  of  the  parlicle  begins, 
the  initial  velocity  and  apace  are  zero,  and  (1)  becomes 

ii^  =  gy.  (3) 

This  shows  that  the  velocity  of  tiio  particle  at  any  time 
is  entirely  indeiwndent  of  the  form  of  the  curve  on  which 
it  moves;  and  depends  solely  on  the  perpendicular  distance 
through  which  it  falls. 

193.  Motion  on  a  Circular  Arc  in  a  Vertical 
Plane. — Take  the  vertical  <liameter  as  axis  of  y.  and  its 
lower  extremity  as  origin  ;  then  tlie  ci[Uation  of  I  he  circle  is 


(1) 


a^zzz 

Uy 

-f\ 

dx 

a  ■— 

y  ~ 

'hi- 

X 

(Is 
a 

Action  of 

iag  on  till! 
c  axis  of  1/ 
tiio 


(1) 

e  time  /,. 
af  Art.  190, 


(2) 

clo  begins, 
iconics 

(3) 

b  any  tiiiio 
c  on  whicii 
lar  distance 


Vertical 

//.  and  its 
IJK'  circlo  is 


(1) 


MOTION  ON  A    CIRCULAR  ARC. 


Let  {k,  h)  be  the  point  K  where 
the  particle  starts  from  rest,  and  (x,  y) 
the  point  P  where  it  is  at  the  time  /. 
Tlien  the  particle  will  have  fallen 
through  the  height  HM  —  h  —  y, 
and  hence  from  (3)  of  Art.  198  we 
have 

ds 


351 


o 
Fi8.84  4- 


dt 


=  V  =  V'ig  (/t  —  y)' 


(2) 


Hence  the  velocity  is  a  minimum  when  y  =  h,  and  a 
maximum  when  y  =  0;  and  this  maximum  velocity  will 
carry  the  particle  through  0  to  A''  at  the  distance  h  above 
tlie  liorizontal  line  through  0. 

To  find  the  time  occupied  by  the  particle  in  its  descent 
from  A'  to  the  lowest  point,  0,  we  have  from  (3) 


di  ~  - 


d.'i 


V2y  {h  -  y) 
—  ady 


V2g{h-y)Ci'<ry"~y^) 


by(l)     i'i) 


the  negative  sign  being  taken  since  t  is  a  decreasing  func- 
tion of  .S'. 

This  expression  does  not  admit  of  integration  ;  it  may  bo 
reduced  to  an  elliptic  iti/eyral  of  the  first  kind,  and  tables 
are  giveiv  of  the  approximate  values  of  the  integral  for 
given  values  of//.* 

If,  however,  the  radius  of  the  circle  is  large,  and  the 
<Te!it(v<t  distance  fCO.  over  wiiich  the  particle  moves,  is 
small,  we  may  dcelupe  ('•))  into  a-  series  of  terms  in  ascend- 
in"  powers  of  ■    ,  and  thus  find  the  integral  ap])roximatelv, 

■■^  '  'iu 


Sue  Ijegendro'is  Traltd  dee  Foiictloiin  Elllptl(iiic». 


352 


THE  SIMPLE  PESDVLUM. 


Let  T\)Q  the  time  of  motion  of  the  ])article  from  A' to  K\ 
L  e.,  from  y  —  h,  through  y  =  0,  to  y  =  h  again,  tlien  (3) 
becomes 


V  !J ^h  y/hy  —  y^^        ^^^' 
integrating  each  term  separately  we  have 


-=^\/;DH-«.^o'(^y 


+  (2T4- «)  (2-;.)  +  '"'■ 


(4) 


which  is  the  complolc  expression  for  t!;e  time  of  moving 
'rom  the  extreme  position  A' on  one  side  of  the  vertical  to 
the  extreme  jKisition   A"  on  the  other;  this  Is  called  an 
oscillation.     (See  Price's  Anal.  Mechs.,  Vol.  1.,  p.  518.) 
If  the  arc  is  very  small,  //  is  very  small  in  comparison 

with  «,  and  ail  the  terms  containing  ^~  will  be  very  small, 

and  by  neglecting  them  (i)  becomes 


-V^- 


ip) 


194.  The  Simple  Pendulum. — Instead  of  supposing 
the  particle  to  move  on  a  curve,  wo  may  imagine  it  sus- 
])fniled  bv  a  siring  (tf  invarial>le  length,  or  a  thin  rod 
(■(•Msulered  of  no  weight,  mid  moving  in  a  vertical  plane 
about  tiie  point  V :  for.  whether  tin-  force  acting  on  the 
particle  be  the  reaction  of  I  he  curve  or  the  tension  of  the 
string,  its  iuh'usily  is  the  same,  while  its  direct wn,  iu 
either  ca.se  is  along  the  nonnal  to  the  curvy. 


RELATIOS   OF  TIME.   LENGTH,   ETC. 


353 


n  K  to  K\ 

w,  then  (3) 


(4) 

of  moviiif!; 

vertical  (o 
I  called  ail 
).  518.) 
comiiarisoii 

very  sniull, 


(6) 


supposing 
^inc  it  Hus- 
a  thin  rod 
•tical  plane 
ing  on  the 
ision  (if  the 
'ii'cclion,  iii 


When  tho  particle  is  supposed  to  be  suspended  by  a 
bread  without  weight,  it  boconies  what  is  termed  a  miiple 
ptKduluw  ■  t'^'d  although  such  au  instrument  can  never  be 
pertLctly  attained,  but  exists  only  in  theory,  yet  approxima- 
tions may  be  made  ^o  it  sufHciently  near  for  practical  pur- 
jwses,  aiid  by  met-iis  of  Dynamics  we  may  reduce  the 
calculation  of  the  niotion  of  such  a  pendulum  to  that  of 
the  simple  pendulum. 

If  I  is  the  length  of  the  rod,  the  time  of  an  oscillation  is 
approximately  given  by  the  formula 


—V^ 


(1) 


when  tho  angle  of  oscillation  is  very  small,  /.  e.,  not  ex- 
ceeding about  4° ;  *  and  therefore,  for  all  angles  between 
this  and  zero,  the  times  of  oscillation  of  the  same  pen- 
dulum will  not  perceptibly  differ ;  /.  e.,  in  very  small  arcs 
thfl  osrillalions  may  be  regarded  as  isochronal,  or  a^j  all 
performed  in  the  same  time. 

195.    Relation   of  Time,    Length,  and    Force    of 

Gravity.— From  (I)  of  Art.  l'J4,  we  have  Tec  V/ if  i/ i« 
constant;  T<x.    ---  if  /  is  constant;  <j<x.l\i  7' is  constant, 

that  is 

(1)  Vov  i\\c  rnma  \t\aGQ  the  times  of  nsciUalion  are  as  the 
square  roots  of  the  lengths  of  tfie  pendulums. 

(2)  For  the  same  i)ciululum  the  times  of  osrillation  are 
inversely  as  the  square  rootf  of  the  forre  of  gravity  at 
different  places. 

*  If  the  liiliiRl  IncUiiBlion  t»  r.*,  tlie  Bcconti  term  of  (4)  U  only  0.00047(1 ;  If  V  Ihc 
socoud  term  Is  uul;  O.OOUOlb. 


354  HEIGHT  OF  MOV  STAIN  DETERMINED. 

(3)  For  the  same  time  ike  lengths  of  pendulums  vary  as 
f he  force  of  gravity . 

Hence  by  means  of  tlie  pendulum  the  force  of  gravity  at 
different  places  of  the  earth's  surface  may  be  determined. 
Let  /.  be  the  length  of  a  pendulum  which  vibrates  seconds 
at  the  place  where  t.  -a  value  of  g  is  to  be  found  ;  tlien  from 
(1)  of  Art.  194  we  have 

i  =  ^y^f;  •■■  ^  =  "'^5  (1) 

and  from  this  formula  g  has  been  calculated  at  many  places 
on  the  earth.  The  method  of  determining  L  accurately 
will  be  investigated  in  Chap.  VII. 

Cor. — If  u  be  the  number  of  vibrations  performed  dur- 
ing iV^  seconds,  and  T'the  time  of  one  vibration, 

then  n  =  ~,  by  (1)  of  Art.  194  =  -\/f  •         (2) 

Since  gravity  decreases  according  to  a  known  law,  as  we 
ascend  above  the  earth's  suiface.  the  comparison  of  the 
times  of  vibration  of  the  same  pendulum  on  the  top  of  a 
mountain  and  at  its  bai'*'  would  give  approximately  its 
height. 

196.  The  Height  of  a  Mountain  Determined  with 
the  Pendulum. — A  seconds  pciiduliim  is  carried  to  the  top 
of  a  mountain  ;  required  to  fnil  the  height  of  the  mountain 
by  observing  the  change  in  the  time  of  osrilfation. 

Let  /•  be  the  radius  of  the  earth  considered  spherical ;  h 
the  height  of  tlie  mountain  above  tiie  surface;  /  the  length 
of  the  penduhmi  ;  //  and  //'tin.'  values  of  gravity  cm  llie 
earth's  surface,  and  i>t  the  to]»  of  the  mountain  respectively. 
Then  (Art.  174)  we  have 


(1) 


(2) 


HEIGHT  OF  ilOUNTAiy  DETERMINED. 

(>■  +  h\    .    .,  _    91^    . 


f=(^-)--^' 


{r  +  hy 


355 


(1) 


which  is  the  force  of  gravity  at  the  top  of  the  mountain. 

Let  «-  =  the  number  of  oscillations  which  the  seconds 
pendulum  at  the  top  of  the  mountain  makes  in  24  hours; 

24  X  60  X  CO 


then  the  time  of  oscillation  = 
(i)  of  Art.  195,  we  have 

24  X  60  X  60 


Hence  from 


71 


n 


h 
r 


24  X  60  X  60 


n 


-  1,  (since  ttW- =  1),  (2) 


which  gives  the  height  of  the  mountain  in  terms  of  the 
radius  of  the  earth.  For  the  sake  of  an  ex!>  „;'\  suppose 
tlie  ponduliim  to  lose  5  seconds  in  a  day  ;  t!.  w  '-  to  make 
5  oscillations  less  than  it  would  make  on  the  su.i..  .;€  of  the 
earth. 

Then  /t  =  24  x  60  x  60  -  5 ; 

which  in  (2)  gives 

h  _      24  X  60  X  60 
r  —  24  X  CO  X  60  —  5 

»  1 


-1 


~V        24  X  60  X  12/         ^ 


24  X  60  X  12 


-  nearly ; 


h 


4000 


^.  =  {  miie,  nearly, 


24  X  CO  X  12 

f  being  4000  miles  (apjn-oximately). 

197.  The  Depth  of  a  Mine  Detennined  by  Ob- 
serving the  Change  of  Oscillation  in  a  Seconds 
Pendulum. — Let  /•  be  the  radius  of  the  earth  as  in  the 


;55(j  CENTRU'ETAL    FORCE. 

last  ciiso  ;  //  tlio  depth  of  the  niiiu-  ;  (j  and  <j  the  vuliics  of 
gravity  on  tlie  earth's  surlaee  and  at  the  bottom  of  the 
mine.     Then  (Art.  171)  we  have 

g'  ~  r  -h  ''^' 

Let  )i  —  the  number  of  oscillations  which  the  seconds 
pendulum  at  tiie  bottom  of  the  mine  makes  in  24  hours. 

,p,  24  X  ()0  X  00  /       Tr 

=  \/rI-A- 

^  _  /j  _  / n \2 

•'•     ^     ■  r  —  \24  X  00  X  GO/' 

from  which  h  can  be  found.  If,  as  before,  the  pendulum 
loses  5  seconds  a  day,  we  have 


h 
r 


1  _  (i I y 

V         24  X  GO  X  12/ 


nearly, 


~  13  X  GO  X  12 

.  • .    /<  =  ^  mile  nearly. 

(See  Price's  Anal.  Mech's,  Vol.  I,  p.  590,  also  Pratt's 
Mech's,  p.  ;J7G.) 

198.  Centripetal   and   Centrifugal  Forces. — Since 

the  pressure  -,  at  any  j)oint,  depends  entirely  upon  the 

velocity  at  that  poilit  and  the  radius  of  curvature,  it  would 
remain  the  same  if  the  forces  X  and  Y  were  1)oth  zero,  in 
which   case   it   would   be   the    wliole  normal  pressure,   /»', 


A 


CEXTKIFVCAL   FORCE. 


357 


\  ill  IK'S   of 

torn  of  tlif 


(1) 

lio  seconds 
4  hours. 


penduliiin 


Jso  Pnilt's 

es. — Since 

upon  flic 

',  it  W(uild 
Ml  zero,  ill 
ossure,  li, 


0 


against  the  curve  It  is  easily  seen,  therefore,  that  this 
pressure  arises  entirely  from  the  incrlia  of  the  moving 
particle,  /.  e.,  from  its  tendency  at  any  point,  to  move  in 
the  direction  of  a  tangent ;  and  this  tendency  to  motion 
along  the  tangent  necessarily  causes  it  to  exert  a  prest^ure 
against  the  deflecting  curve,  and  which  requires  the  curve 

to  oi)pose  the  resistance  —  •     Hence,  since  tlio  particle  if 

left  to  itself,  or  if  left  to  tlie  action  of  a  force  along  the  tan- 
gent, would,  by  th  '  law  of  inertia,  continue  to  move  along 

that  tangent,  —  ia  the  effect  of  tlie  force  whicii  deflects  the 

liarticle  from  its  otherwise  rectilinear  path,  and  draws  it 
towards  tlie  cen''''o  of  curvature.  This  force  is  called  the 
Ceiifrijietal  Force,  which,  therefore,  may  be  dtfiued  to  be 
f/if  force  which  deflects  a  particle  from  its  otherwise  recti- 
linear path.  Tiie  equal  and  opposite  reaction  exerted  away 
from  the  centre  is  called  the  CenlrifiKjal  Force,  which  may 
he  defined  to  he  the  resistance  ivhich  the  inertia  of  a  particle 
in  motion  opposes  to  'whatever  deflects  it  from  its  rectilinear 
path.  Centripetal  and  centrifngal  are  therefore  tlie  same 
<iuantity  under  different  as-pects.  The  action  of  the  former 
is  towards  the  centre  of  curvature,  while  that  of  the  latter 
i>i  from  the  centre  of  curvature.  The  two  arc  called  central 
forces.  They  determine  the  direction  of  motion  of  the  par- 
ticle hut  do  not  affect  tlie  velocity,  since  they  act  continu- 
ally at  right  angles  to  its  path.  If  a  particle,  attached  to  a 
string,  be  whirled  abo,  t  a  centre,  the  intensity  of  these 
central  forces  is  measured  by  the  tension  of  the  string.  If 
tiie  string  be  cut,  the  piirticle  will  move  along  a  tangent  to 
the  curve  with  unchanged  velocity. 

Coit.    1.' — If  m  be  the  mass  moving  with  velocity  v,  its 
cenlrifugiil   force   is  m  -•      If   w  be   tlie  anguhir  velocity 


358 


CEXTlilFiaAL    FORCE. 


described  by  the  radius  of  curvature,  thcu  (Art.  1G(»,  Ex.  1), 
V  =  ou),  and  consequently 


the  centrifugal  Ibrco  of  in  =  rru^p. 


(1) 


Cor.  2. — Let  m  move  in  a  circle  with  a  constant  velocity, 
v;  let  a  =  the  radius  of  the  circle,  and   2'  the  time  of  a^ 
complete  revolution  ;  then  iTra  =  vT; 

.-.     the  centrifugal  force  oim  =  m  -^  ;  (2) 

ai>d  thus  the  centrifugal  force  in  a  circle  varies  directly  as 
the  radius  of  the  circle,  and  invert^ely  as  (he  square  of  the 
pp.riadic  tini  . 

Cor.  3.— If  m   moves  in   the   circle    with    a    constant 
angular  velocity,  u>,  then  (Art.  1(50,  Ex.  1),  v  =  aw ; 


the  centrifugal  force  oi  in  =  md^( 


(3) 


and  therefore  varies  directly  as  the  radius  of  the  circle. 

Thus  if  a  i)article  of  mass  ni  is  fastened  by  a  string  of 
length  a  to  a  point  in  a  horizontal  plane,  and  describes  a 
circlt.  ni  the  plane  about  the  given  point  as  centre,  the  cen- 
trifugal force  produces  a  tension  of  the  string,  and  if  w  is 
the  constant  angular  velocity,  the  tension  =  m  u^a. 

199.  The  Centrifugal  Force  at  the  Equator.— Let 

Ji  denote  tiie  eciuatorial  radius  of  the  earth  =  2092G202* 
feet,  T  the  time  of  revolution  upon  its  axis  —  80104 
seconds,  and  tt  =  3.1415926.  Substituting  these  values  in 
(2)  of  Art.  198,  and  denoting  the  centrifugal  force  at  the 
e()uator  by/,  and  the  mass  by  unity,  we  have 


/  = 


0.1  U2(;  feet. 


(1) 


•  E 


.  Brit.,  Art.  Geodesy. 


CENTRIFVilArj   FORCE. 


359 


G(»,Ex.  1), 


(1) 

t  velocit}', 
time  of  a 


(2) 

liredly  as 
are  of  the 

constant 
fo) ; 

(3) 

circle. 
[I  string  of 
lcscriV)es  a 
e,  the  cen- 
nd  if  w  is 

ator. — Let 

20926^02* 
3  -  80164 
e  values  in 
>rce  at  the 


0) 


The  force  of  gravity  at  the  equator  has  been  found  to  be 
32.09023  ;  if  Hiis  force  were  not  diminished  by  the  cen- 
trifugal force  ;  i.  e.,  if  the  earth  did  not  revolve  "bn  its 
axis  the  force  of  gravity  at  the  e(iuator  would  be 


G  =  32.0!)022  +  0.1112(i  =  32.20U8  feet. 


(•^') 


To  (k'terniine  tlie  relation  between  the  centrifugal  force 
and  the  force  of  gravity,  we  divide  (1)  by  (2)  which  gives 


/  _   0.11126 

O  ~  32.20148  ~  289 


«o>  iiearly. 


(3) 


that  is,  the  centrifiKjal  fortV  at  tlie  equator  in  ^J^  of  that 
which  the  force  of  (/rarity  at  the  equator  would  be  if  the 
earth  did  not  rotate. 


200.  Centrifugal  Force  at  Differ- 
ent Latitudes  on  the  Earth.— Let 

P  be  any  particle  on  the  earth's  surface 
describing  a  circumference  about  the 
axis,  NS,  with  the  radius  PD.  Ix»t 
0  =  ACP  =  the  latitude  of  P;  R 
the  radius,  AC,  of  the  earth  ;  and  R' 
the  radius  PD  of  the  parallel  of  lati- 
tude passing  through  P.     Then  we  have 

A"  =  R  cos  (p. 


c 

N-^, 

^Y^ 

vj 

jx 

s 
Fig.  85 


(1) 


Let  the  centrifugal  force  at  the  point  P,  which  is  exerted 
in  the  direction  of  the  radins  DP,  be  represented  by  the 
line  FB.  Resolve  this  into  the  two  components  PF,  act- 
ing along  the  tangent,  and  PB,  acting  along  the  normal. 
Tlien  by  (2)  of  Art.  198  we  have 


PB  =  ^"^'L 


in^R  cos  (ft 


,  by  (1). 


(2) 


3U0 


CEyTh-IFIdAL   FOliCE. 


Hence,  the  crnh'tfitc/al  forrv  (it  nini  point  on  tJiv  earth's 
fnirface  varies  directly  as  tlie  rosine  of  the  latitude  of  the 
place. 

For  the  normal  coinj)()ncnt  we  have 

PE  =  PB  cos  0 
4Tr^/i*  cos^  (\) 


T-f 


by  (2) 


=  /cos2  (/),  hy  (1)  of  Art.  1!)!».    (:{) 

Hence,  the  component  of  the  centrifugal  force  which  direrth/ 
opposes  tin:  force  (f(/rarifi/,  at  ainj  point  on  the  earth's  sur- 
face, is  equal  to  tliv  coitrifuijal  force  at  the  equator,  viul- 
ti plied  by  tite  square  of  tlie  cosine  of  the  latitude  of  the 
phCce. 


Also 


/'/'  =  PB  sin  f/. 

in'^li  sin  (/)  cos  0 


,  by  {'i) 


=  {,m\  2(p,  by  (1)  of  Art.  1!)9  ;  (4) 

that  is,  the  component  of  Hic  centrifiiyal  force  which  tends 
to  draw  jiart ides  from  auy  parallel  (f  latitude,  P,  towards 
the  equator,  and  to  cause  the  carf'  to  assume  the  figure 
of  an,  oblate  spheroid,  varies  as  tlie  sine  of  twice  the 
latitude. 

T'  proeedinir  calciiliitioii  is  mnde  on  the  liypothesis  tir„.t; 
tile  eartli  is  .i  pert'eet  sphere,  wiiereas  it  is  an  eljlate 
spheroid;  and  llie  attraction  of  tlie  eartii  on  particles  at 
its  snrface  decrease.'  as  we  [>ass  from  tiie  poles  to  the 
eqiuitor.      The    pendulum    furnislies    the    mo.>t    aecuiate 


Uiv  earth's 
iliide  of  the 


.rt.  19!).   (3) 

li ich  dircilhj 
■  eartli'x  -sur- 
juafor,  mul- 
'ilude  of  the 


irt.  109  ;  (4) 

wJiicli  tends 
",  P,  towards 
nc  the  figure 
f   twice    the 


'pothesis  tlr.'t; 
is  an  c'Matij 
II  |KirtioU's  at 

polos     to    till' 

lust    acoiiiato 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


I.I 


■-  IIIIIM 


Mi 


u 


IIIIIM 

II  2.2 
\= 

1.8 


1.25      1.4 

1.6 

^ 6"     — 

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ri'E  rO.MCAL   I'K.\Dri.UM. 


301 


a 


metliod   of  dolenniiiiiig   the  force  of  gravity  ut  tliffereul 
l)lu'3os  on  tlic  eartli'i^  surface. 

201.  The  Conical  Pendu- 
lum.— The  Governor. — Suppose 
a  parucle,  /'.  of  luast;  w,  to  be  at- 
tached to  one  end  of  a  string  of 
length  /,  tlie  ctlier  end  of  which  is 
fixed  at  A.  Tlie  particle  is  made 
to  describe  a  horizontal  circle  of 
radius  PO,  with  uniform  velocity 
round  the  vertical  axis  .KAso  that 
it  makes  n  revolutions  per  second. 
It  is  required  to  find  the  inclina- 
tion, 0,  of  the  string  to  the  vertical, 
and  the  tension  of  the  string. 

The  velocity  ')1  P  in  feet  per  second  =  'iirn-OP  —  2mi  I 
sin  0.  The  forces  acting  upon  it  arc  the  tension,  7',  of  the 
string,  the  weight,  w.  of  the  particle,  aiul  the  centrifugal 

force,  m  '^i'-f  f^*^—  (Art.  1 1)8).     1  lence  resolving,  we  have 
'  I  sm  0 


Fi9.86 


7' COS  0   =z    DIIJ. 

in-  irt'^n^  I, 


(a) 
(3) 

(4) 


for  horizontal  forces.     7'siu  0  —  in-\"'hiU  sin  0  ;     (1) 

for  vertical  forces, 

Fnmi  (1)  T 

which  in  (5i)  gives 

where  7' and  0  are  emnpletcly  determined. 

if  the  string  be  rejdaced  by  a  rigiil  rod.  which  can  turn 
about  .1  in  a  ball  and  socket  joint,  the  instrument  is  called 
a  conical  pendulum,  autl   occurs   in    the  (joucmor  of  the 
steum-engine. 
10 


m 


3G2 


EXAMPLES. 


EXAMPLES. 

1.  If  the  length  of  the  .seconds  iiendulnm  he  39.1393 
inche.s  in  London,  lind  tlie  value  of  y  to  three  places  of 
I'fcinials.  Am.  32.191  feet. 

2.  In  what  time  will  a  pendulum  viln-ate  whose  length  is 
15  ineiies  ?  ,i„,,.  0.62  sec.  nearly. 

3.  In  what  time  will  a  pendulum  vibrate,  whose  length  is 
double  that  of  a  seconds  pendulum  ?  Ans.  1,41  sees. 

4.  How  iiuiny  vibrations  will  a  pendulum  3  feet  long 
make  in  a  minute?  Ans.   02.55. 

5.  A  peiululum  which  beats  seconds,  is  taken  to  the  top 
of  a  mountain  one  mile  high  :  it  is  re(|uired  to  find  the 
number  of  seconds  which  it  will  lose  in  12  hours,  allowing 
the  radius  of  the  earth  to  be  4000  miles.     Ahk.   10.8  sees. 

(i.  What  is  the  length  of  a  pendulum  to  beiit  seconds  at 
the  place  where  a  body  falls  lOjV  ft.  in  the  first  second  ? 

Ahk.  39,11  ins.  nearly. 

7.  If  39.11  ins.  be  taken  as  the  length  of  the  seconds 
pendulum,  how  long  must  a  pendulum  be  to  beat  10  times 
in  a  minute?  Ans.  11  Tift. 

S.  A  particle  slides  down  the  arc  of  a  circle  to  the 
lowest  point;  lind  t!ie  velocity  at  the  lowest  point,  if  the 
angle  described  round  the  centre  is  'iO  .  Am.    \/gr. 

9.  A  pendulum  which  oscillates  in  a  second  at  one  place, 
is  carried  to  another  place  where  it  makes  120  more  oscil- 
lations in  a  day :  lomjiare  the  force  of  gravity  at  the  latter 
place  with  that  at  the  ibrnier.  Ans.   (ll\l)'K 

10.  Find  the  nnmliei'of  vihral  ions. ;/,  .which  a  pendiiliiin 
will  gain  in  .V  si!coiids  by  shortening  the  length  of  the 
pendulum. 


nm  1)0  39.1393 
tliroo  places  of 
32.191  feel. 

ivhose  length  is 
'i  sec.  ne.irly. 

whose  leiigtii  is 
'IS.  1,41  sees. 

m   3  feet  long 
Ahs.   G2.55. 

ken  to  the  top 
•ed  h)  lind  (lie 
loiirs,  allowing 
w.   10.8  sees. 

)eat  seconds  at 
rst  second  ? 
I  ins.  nearly, 

of  (lie  seconds 
I  beat  10  times 

ins.  iir^ft. 

circle    to  the 
st  point,  if  the 
Ans.    Vgr. 

d  at  one  place, 
20  more  oscii- 
\-  at    the  latter 

"■"■  mny- 

'h  a  pendnhim 
length  v\'  the 


< 


EXAMPLES. 


363 


Let    the   length,   I,   he   decreased    hy   a   small   (piantity, 
/,.  and  let  n  bo  increased  by  ?«i  ;  then  from  (2)  of  Art.  195 


we  get 


which,  divided  by  (2)  of  Art.  195,  gives 
Hence 


11.  If  a  pendulnm  be  i')  inches  long,  how  many  vibra- 
tions will  it  gain  in  (mo  day  if  the  bob*  be  screwed  np  one 

tnrn,  the  screw  having  32  threads  to  the  inch  ? 

.'l/(.s.   28. 

12.  If  acloek  loses  two  niinuto  a  day,  how  many  tnrns 
to  the  right  hand  must  we  give  the  nut  in  order  to  correct 
its  error,  su^jposing  the  screw  to  iiave  oO  threads  to  the 
iiicl,  y  Ans.  5-4  turns. 

13.  A  mean  solar  day  contains  24  hours,  3  minutes, 
50-5  seconds,  sidereal  time  ;  calculate  the  luiigth  of  the 
pendulum  of  a  clock  beating  sidereal  seconds  in  London. 
Sw  Ex.  1.  ''"*'■  38-935  inches. 

14.  A  lieavy  ball,  suspended  by  a  fine  wire,  vibrates  in  a 
small  arc;  48  vibrations  are  counted  in  3  niinuti's.  Cal- 
culate the  length  of  the  wire.  Alls.   45-87  feet. 

15.  The  height  of  tiic  cupola  of  St.  Paul's,  aliove  the 
floor,  is  340  ft.;  calculate  tiie  nunilier  ol'  vilirations  a  heavy 
body  would  mnke  in  iialf  an  hour,  if  suspended  from  the 
dome  by  a  line  wire  wliieli  reaches  to  wilhiu  H  inches  of 
the  floor.  •I""-   K'i-l- 


*  The  lowor  oxtrenilly  of  tho  iiumlulum. 


mm 


364 


EXAMPLES. 


IG.  A  seconds  pendulum  is  carried  to  the  top  of  a 
mountain  m  niiles  liigli  ;  assuming  that  the  force  of 
gravity  varies  inversely  as  tiie  square  of  the  distance  from 
the  centre  of  the  earth,  find  the  time  of  an  oscillation. 


Ans. 


/4000  +  m. 


40UU 


)" 


sees. 


17.  Prove  that  the  lengths  of  jicndnlums  vibrating  dur- 
ing the  same  time  at  the  same  i)lace  are  inversely  as  the 
squares  of  the  number  of  oscillations. 

KS.   In  a  series  of  experiments  made  at  Ilarton  coal-pit,  a 

pendulum  which    beat   seconds  at  the  surface,  gained  -i\ 

beats  in  a  day  at  a  tlepth  of  12G0  ft.;  if  //  and//'  be  the 

force  of  gravity  at  the  surface  and  at  the  depth  mentioned, 

show  that 

9'  -  0  _       1 
— ^      —  Tff^irO' 

lit.  A  pendulum  is  found  to  make  fi-tO  vibrations  at  the 
e(|uator  in  the  same  time  that  it  makes  041  at  Greenwich; 
if  a  string  inuiging  vertically  can  just  sustain  80  lbs.  at 
(Jreenwich,  how  nuuiy  lbs.  can  the  same  string  sustain  at 
the  equator?  Ans.  80|- lbs.  about. 

'v'O.  Find  the  time  of  descent  of  a  jjarticle  down  the  arc 
of  a  cychiid,  tiu'  axis  of  the  cycloid  lu'ing  vertical  and  vertex 
downward  ;  and  show  that  the  time  of  descent  to  the  lowest 
point  is  the  same  whatever  point  of  the  curve  the  particle 

starts  from.  ^  ' r 

A  us.  Ti  \  /    • 

'.M.  If  in  Ex.  '^0  the  ])articl(>  begins  to  move  from  the 
extremity  of  (hf  liasi'  ol'  tlic  cycloid  linil  the  pressure  at  tlie 
lowest  point  of  the  curve. 

Ans.  'i(/\  i.  r.,  the  ])ressur(!  is  twice  the  weight  of  the 
particle 


the   top  of  a 
tlio    force    of 
listiiiK'c  from 
HC'illatioii. 


IJOO 


')' 


sees. 


vil)riiting  dur- 
iverselv  a.s  the 


rtoii  coal-pit,  a 
ICC,  guineil  2^ 
I  iiiul  (f  he  the 
ith  mentioned, 


irations  at  tlie 
at  (ireenwich  ; 
tain  80  li)s.  at 
ring  sustain  at 
^\  Ibt?.  about. 

e  down  tlic  arc 
tieal  and  vertex 
it  to  the  lowest 
I've  the  particle 

inn.   r,  \  /     . 
V   // 

move  IVimi  tlie 
pressure  at  t!io 

weight  ul'  the 


KX.iMI'hkS. 


'jf^n 


Hfi/ 


22.  Find  the  pressure  on  the  lowest  point  of  the  curve 
in  Art.  19.'],  (I)  when  the  i)article  starts  from  rest  at  the 
highest  point.  A.  (Fig.  S4),  {'i)  when  it  starts  from  rest  at 
the  point  B. 

A)is.  (1)  0^;  (2)  ;i// ;  i.e.,  (1)  tiie  pressure  isfivctinus 
tlie  weight  of  the  particle  and  (2)  it  is  three  times  tlie 
weight  of  the  particle. 

215.  Tn  the  simple  itendulum  find  the  point  at  which  the 
tension  on  the  string  is  the  same  as  when  the  jiarticle 
liangs  at  rest. 

Anx.  y  =  §//,  where  //  is  the  height  from  which  the 
pendulum  has  fallen. 

24.  If  a  particle  be  compelled  to  move  in  a  circle  with  a 
velocity  of  300  yards  per  minute,  the  radius  of  the  circle 
being  IG  ft.,  find  the  centrifugal  ferce. 

Ans.   14- OG  ft.  per  sec. 

25.  If  a  body,  weighing  IT  tons,  move  on  the  circum- 
ference of  a  circle,  whose  radius  is  1110  ft.,  with  a  velocity 
of  K)  ft.  per  sec,  find  the  centrifugal  force  in  tons  (take 
g  =  32-1948).  Ans.  0-1217  ton. 

2G.  If  a  body,  weighing  1000  lbs.,  be  constrained  to  move 
in  a  circle,  whose  radius  is  100  ft.,  by  means  of  a  string 
capable  of  sustaining  a  strain  not  exceeding  450  lbs.,  find 
the  velocity  at  the  moment  the  string  lireaks. 

Ans.  38.06  ft.  per  sec. 

27.  If  a  railway  carriage,  weighing  7-21  tons,  moving  at 
the  rate  of  30  miles  per  hour,  describe  a  portion  of  a  circle 
whose  radius  is  4G0  yards,  find  its  centrifugal  force  in  tons. 

Ans.  0-314  ton. 

28.  If  the  centrifugal  force,  in  a  circle  of  100  ft.  radius, 
be  140  ft.  per  sec.,  I'.id  the  periodic  time. 

Ans,  5-2  sees. 


;{co 


EXAMPLES. 


29.  If  the  centrifugal  force  lie  i:U  ozs.,  and  tlio  ratlins 
of  the  circle  100  ft.,  the  iieriodic  time  being  one  hour,  find 
the  weight  of  the  Ixuly.  An.s.  :]8G-'3m  ioufi. 

30.  P^ind  the  force  towards  the  centre  re(|iiired  to  make 
a  body  move  uniformly  in  a  circle  whose  radins  is  5  ft., 
with  .such  a  velocity  as  to  coniplete  a  revolution  in  5  sees. 

Ans.  -— • 
5 

.11.  A  stone  of  one  lb.  weight  is  whirled  round  horizon- 
tally by  a  string  two  yards  long  having  one  end  fixed  ;  find 
the  time  of  revolution  when  the  tension  of  the  string  is  3  lbs. 


Ans.  2 


^^] 


sees. 


32.  A  weight,  w,  is  placed  on  a  horizontal  har,  OA, 
which  is  made  to  revolve  round  a  vertical  axis  at  0,  with 
the  angular  velocity  6);  it  is  re(iuired  to  determine  the 
position,  A,  of  the  weight,  when  it  is  upon  -the  i)oint  of 
sliding,  the  coefficient  of  friction  being  /'. 

fg 


Ans.  OA  z= 


w*< 


33.  Find  the  diminution  of  gravity  at  the  Sun's  equator 
caused  by  the  centrifugal  force,  the  radius  of  the  Sun  being 
441000  miles,  and  the  time  of  revolution  on  his  axis  being 
G07  h.  48  m.  J«.v.  0-  0192  ft.  per  sec. 

34.  Find  the  centrifugal  force  at  the  equator  of  Mercury, 
the  radius  being  1570  miles,  and  tiie  time  of  revolution 
■«4  h.  .')  m.  Ans.  0.0435  ft.  per  sec. 

35.  Find  the  centrifugal  force  at  the  equator,  (1)  of 
V^enus,  radius  being  ;59()0  miles  and  rime  of  revolution 
23  h.  21  m.,  (2)  of  Mars,  radius  being  2050  miles  and 
l)eriodic  time  24  h.  37  m.,  (3)  of  Juiuter,  radius  being 
43500  miles  and  periodic  time  9  h.  50  m.,  and  (4)  of  Saturn, 
radius  })eing  395H0  miles  and  periodic  time  10  h.  29  m. 


[iikI  the  radius 
one  hour,  tiiul 
80- 309  toiif^. 

iiircd  to  make 

radius  is  5  ft., 

ion  in  5  sees. 

4Tr2 
Ans.  —-• 
5 

round  horizon- 
ind  tixed ;  lind 
I  string  is  3  lbs. 


VI 


sees. 


ntal  bar,  OA, 
axiti  at  0,  with 
di'tcrmine  the 
I  -the  point  of 


OA 


Sun's  equator 
the  Sun  being 
.  his  axis  being 
I  ft.  per  sec. 

or  of  Mercury, 
of  revolution 
5  ft.  per  sec. 

quator,  (1)  of 
of  revolution 

)50  miles  and 
radius  being 

I  (4)  of  Saturn, 

)  h.  29  m. 


KXAMl'LES. 


.Jfi  1 


Aiif.  (1)  0-11504  ft.  per  sec.;  (2)  0-0544  ft.  per  sec; 
(;i)  7- 0907  ft.  per  sec;  (4)  5-  7924  ft.  per  sec 

3fi.  Find  tlie  eft'ect  of  centri<"ugal  force  in  diniinisUin^ 
Ifravity  in  the  latitude  of  t]0  .     [See  (3)  of  Art.  200). 

J  lis.  0-028  ft.  per  sec. 

37.  Find  (1)  the  diniinutio!!  of  gravity  caused  by  cen- 
trifugal force,  and  (2)  the  component  wliich  urges  j)articles 
towards  the  equator,  at  the  latitude  of  23°. 

Ans.   (1)  0-09  ft,  per  sec;  (2)  0-04  ft.  pr  sec, 

38.  A  railway  carriage,  weighing  12  tons,  is  moving 
along  a  circle  of  radius  720  yards,  at  the  rate  of  32  miles 
an  hour;  find  the  horizontal  pressure  on  the  rails. 

Ans.  0-38  ton,  nearly. 

39.  A  railway  train  is  going  smoothly  along  a  curve  of 
500  yards  radius  at  the  rate  of  30  miles  an  hour;  find  at 
what  angle  a  pluml)-line  hanging  in  one  of  the  carriages 
will  be  inclined  to  the  vertical.  Ans.  2°  18'  nearly. 

40.  The  attractive  force  of  a  mountain  horizontally  is/ 
and  the  force  of  gravity  isg;  show  that  the  time  of  vibra- 

tion  of  a  pendulum  will  be  "W,^^?^'  "  '^^'"^  ^^^^  length 
of  the  pendulum. 

41.  In  motion  of  a  particle  down  a  cycloid  prove  that  the 
vertical  velocity  is  greatest  when  it  has  completed  iialf  its 
vertical  descent. 

42.  When  a  particle  falls  from  the  highest  to  the  lowest 
point  of  a  cycloid  sliow  that  it  describes  half  the  path  in 
two-thirdr  of  the  time. 

43.  A  railway  train  is  moving  smoothly  along  a  curve  at 
the  rate  of  GO  miles  an  hour,  and  in  one  of  the  carriages  a 
pendulum,  which  would  ordinarily  oscillate  seconds,  is 
observed  to  oscillate  121  times  in  two  minutes.  Show  that 
the  radius  of  the  cui've  is  very  nearly  a  (piarter  of  a  mile. 


m 


:{cs 


KXMfPl.RS. 


44.  One  end  of  a  string  is  fixed  ;  to  tlie  otlier  end  a 
particle  is  attached  wliich  describes  a  horizontal  circle  with 
iinirorni  velocity  so  tliut  the  string  is  always  iclined  at  an 
angle  of  GO  to  the  vertical;  show  that  the  velocity  of  the 
partiele  is  tiiat  which  would  he  acquired  in  falling  freely 
from  rest  tlirough  a  space  equal  to  three-fourths  of  the 
length  of  tiie  string. 

45.  The  horizontal  attraction  of  a  mountain  on  a  particle 
At  a  certain  place  is  such  as  would  produce  in  it  un  accelera- 
tion denoted  hv    •     Show  that  a  seconds  pendulum  at  that 

,  ...      .     21000.      .    .        ,  , 

place  wul  gain    — j—  beats  in  a  day,  very  nearly. 

46.  In  Art.  201,  sujjpose  I  eijiial  to  2  ft.  and  m  to  be  20 
lbs.,  and  that  the  system  makes  10  revolutions  per  sec,  and 
g  =  ;32;  find  0  and  T. 

Ans.  0  =  co.s-i  -^  ,;  T  =  5007t2  pounds. 

25Tr'^ 

47.  A  tube,  bent  into  the  form  of  a  plane  curve,  revolves 
with  a  given  angular  velocity,  about  its  vertical  axis;  it  is 
required  to  determine  the  form  of  the  tube,  when  a  heavy 
particle  placed  in  it  remains  at  rest  in  all  parts  of  the 
tube. 

(Take  the  vertical  axis  for  the  axis  of  y,  and  the  axis  of  x 
horizontal,  and  let  w  =  the  constant  angular  velocity). 
Ans.  x^ui^  =  2gy,  if  a;  =  0  when  i/  =.  0,  i.  e.,  the  curve 
is  a  parabola  whose  axis  is  vertical  and  vertex  downwards. 

48.  A  particle  moves  in  a  smooth  straight  tube  which 
revolves  with  constant  angular  veloeity  round  a  vertical 
axis  to  which  it  is  perpendicular,  to  determine  the  curve 
traced  by  the  particle. 

Let  (.)  =  the  constant  angidar  velocity:  and  (r,  6)  the 
position  of  the  particle  at  tliL-  time  /,  anu  let  r  =:  a  when 


lie  otlier  end  n 
)ntiil  circle  with 
8  iciiiu'd  at  an 
L'  velocity  of  the 
in  falling  IVeely 
'-fourths  of   the 


ain  on  a  particle 
in  it  un  aoc^elcra- 

mduhim  at  that 
!arly. 

,nd  m  to  be  20 
jns  per  sec.,  and 

500tt2  pounds. 

?  curve,  revolves 
^•tical  axis ;  it  is 
,  when  a  heavy 
dl  parts  of   the 

md  the  axis  of  x 
ifjiilar  velocity). 
i.  p.,  the  curve 
ex  downwards. 

ght  tube  which 
•dund  a  vertical 
•mine  the  curve 

;  and  (r,  B)  the 
let  r  =  a  when 


KXAMVLKS. 


300 


/  =  0.      Then   since   the   motion   of    the   particle   is.  due 
entirely  to  the  centrirnga!  force,  wu  have 


dr 


if    -  =  0,  when  r  =  a.     Hence  we  have 
dt 

ii 


m 


chapti:r    iv. 


IMP  A  C  T  . 


202.  An  Impulsive  Force. — llitlierto  \vc  have  con- 
sidered force  only  as  coii/iiiiioKs.  i,  >:,  as  acting  tiirougii  a 
(J"iinite  and  finite  i)ortioii  of  time,  and  jjrodncing  a  finite 
cliaiige  of  velocity  in  tliat  time.  Siicli  a  force  is  measured 
at  any  instant  by  the  mass  on  whicii  it  acts  niiilti})lied  by 
the  acceleration  which  it  causes.  If  a  ]»articleof  nniss  in  be 
moving  with  a  velocity  v,  and  be  retarded  by  a  constant 
force  whicli  Itrings  it  to  rest  in  the  time  /,  then  the  measure 

of  this  force  is  ---  (Art.  'H)).  Now  suppose  the  /i)in'  t  dur- 
ing which  the  particle  is  brouj;ht  forest  to 'le  made  very 
small;  then  the  /o/yv  rcipii red  to  bring  it  to  rest  must  be 
very  large ;  and  if  we  suppose  /  so  small  that  we  are  unable 
to  measure  it,  fiien  the  force  becomes  so  great  tiiat  we  are 
unable  to  oljtain  its  measure.  A  typical  case  is  the  l)low  of 
a  hammer.  Here  the  time  during  which  there  is  contact  is 
apparently  infinitesimal,  certainly  too  small  to  be  measured 
bv  any  ordinary  methods;  yet  the  effect  produced  is  con- 
siderable. Similarly  when  a  cricket  ball  is  driven  back  by 
a  blow  from  a  bat,  the  original  velocity  of  the  ball  is 
destroyed  and  a  new  velocity  generated.  Also  when  a  bul- 
let is  discharged  from  a  gun.  a  large  velocity  is  generated 
in  an  extremely  brief  time.  r\uces  acting  in  this  way  are 
called  impulsive  forces.  An  hnpulsive  force  nnijj  therefore 
he  defined  to  he  a  force  lohich  produces  a  fiiii/e  clianr/e  of 
motion  in  an  indefiniteli/  hrief  time  A  n  Impulse  is  the 
effect  of  a  hlow. 

In  snch    cases  as  these    it   is   impossible    accurately    to 
determine   the   force   and   time;    but    we   can    ileterniinc 


I  \vc  have  con- 
3ting  tlirough  a 
iducing  a  litiito 
rce  is  nioasiired 
ts  imiltiplied  by 
3le  of  rasiss  jn  ha 
1  l)y  a  constant 
It'll  the  measure 

tlie  /i)iH'  t  dur- 

li>. 'lO  made  very 
to  rest  must  be 
it  we  are  uuable 
reat  tluit  we  are 
•e  is  tlie  l)lo\v  of 
ere  is  ooutaet  is 

to  be  measured 
produced  is  eon- 
i  driven  back  l)y 
\-  of  the  ball  is 
Uso  Avhen  a  bul- 
ity  is  generated 

in  this  way  are 
'I'  iiuiy  thcrcfntr 

fiiii/c  cliangp  of 

I  Dip  id  xe  /*■  the 

e    accurately    to 
can    determine 


IMI'Arr  oil    Ct)IJJS/(>\, 


•r,  I 


their  product,  or  Pf,  since  tliis  is  merely  the  cliangc 
in  velocity  caused  by  the  l)low  (Art.  ■>()).  Hence,  in 
tiie  cas^' of  blows,  or  in  Isivc  forces,  we  do  not  attenipt 
to  measiirt'  the  force  ami  tiic  time  of  action  scparati'ly,  but 
simply  take  the  /rjioh'  iniDucnlmn  pvoihunl  ar  ilrshuiju'il,  fis 
llir  nii'itsiirr  nf  the  iin/iiilsc.  lUraiix'  impulsive  forces  pro- 
duce tbcir  rfffi-ts  in  an  indffinilciy  slm.-l  time  they  are 
sometime-!  called  iii.^tfnitdncoiis  forrcs.  I,  c,  forces  requiring 
no  time  for  their  action.  lint  no  >ucli  force  exists  in 
nature:  every  force  requires  tiiiii' hiv  its  action.  There  is 
no  case  in  nature  in  which  a  finite  change  of  motion  is 
])roduced  in  an  infinit.Jmal  of  time  :  for,  whenever  a 
finite  velocity  is  generated  or  destroyed,  a  finite  time  is 
occu])ied  in  the  jjrocess,  though  we  may  be  unable  to 
measure  it.  even  approximately. 

203.  Impact  or  Collision. — When  two  bodies  in  rela- 
tive motion  come  into  contact  witii  each  other,  an   iinpnrt 
or  cotlision  is  .said  to  take  i)lace,  and  i)ressure  begins  to  act 
between  them  to  i)revent  any  of  their  parts  from  jointly 
occupying  the  .^ame  space.     This  force  increases  from  zero, 
when  the  collision  begins,  up  to  a  very  large  magnituile  at 
the  instant  of  greatest  compression.     If,  as  is  always  the 
case  in  nature,  eucii  body  ))osse.s.ses  some  (h'gree  of  elasticity, 
and    if  they  are  not    ke])t    together   after   the    impact   l)y 
cohesion  or  by  some  artificial  means,  the  mutual  pressure 
between  them,  after  reaching  a  maximum,  will  gradually 
diminish    to  zero.      The  whole  ])rocess  woukl  occupy   not 
greatly  more  or  less  than  an  hour  if  th"  bodies  were  of  such 
dimensions  as   the  earth,  and  such   degrees  of  rigidity  as 
copper,  steel,  or  glass.     In   the  case,  however,  of  globes  of 
the.-<e   substances  not   exceeding  a   yard   in   diameter,   the 
whole  i)rocess  is   probably  finished   within  a  thousandth  of 
a  .''econd.* 

Tlie  im]mlsive  forces  are  so  much  more  intense  than  the 

*  Tlinin-ori  iin.i  T;nl's  \at.  Phil.,  p.  274. 


;i:-.' 


DIliKCT  AM)    CHSIHAI.    IMl'ACT. 


ordinary  forces,  that  during  tlic  l)ri('f  linu'  in  wliich  tlio 
i'ornior  act,  an  ordinary  I'oroc  docs  not  prodnce  an  cltVct 
comparahk'  in  anionnt  with  that  })rodnci'd  hy  an  iininilsivc 
force.  For  oxanijiic,  an  iin[iulsivf  force  niight  gi'nerate  a 
velocity  of  1000  in  less  time  tiian  onetcntli  of  a  second. 
while  gravity  in  one-tenth  of  a  second  would  generate  a 
velocity  of  ahoiit  three.  IKiiee.  in  dealing  with  t lie  effects 
of  impulses,  Unite  forces  need  not  he  ccnisidered. 

204.  Direct  and  Central  Impact.  — When  two  hodie8 
impinge  on  each  other,  so  that  their  centres  het'ore  im{)act 
are  moving  in  tiie  same  straigiit  line,  and  the  common  tan- 
gent at  the  i)oint  of  contact  is  perpendicular  to  the  line  of 
motion,  the  impact  is  said  to  he  (linrt  anil  nulnd.  When 
these  conditions  are  not  fidfllled,  the  impact  is  said  to  he 
nbli(iuc. 

When  two  hodies  impinge  diu'ctly,  one  upon  the  other, 
tlie  mutual  action  hetween  them.  a>  any  instant,  nnist  he 
in  the  lino  joining  their  centres;  and  hy  the  third  law 
(Art.  IGG),  it  must  be  e([ual  in  amount  on  the  two  hodies. 
Ilence,  hy  Law  II,  they  must  experience  e([ual  changes  of 
motion  in  contrary  directions. 

We  nuiy  consider  the  impact  as  consisting  of  two  jjarts  ; 
during  the  first  jiart  the  hodies  are  coming  into  closer  con- 
tact Avith  each  other,  mutually  displacing  the  particles  in 
the  vicinity  of  the  point  of  contact,  producing  a  comi)res- 
sion  and  distortion  ahout  tiiat  |)oint.  which  increases  till  it 
reaches  a  nuiximum,  when  the  molecular  rein  i  imis.  thus 
called  into  play,  arc  suiheient  to  resi<i  further  compression 
and  distortion.  At  this  instant  il  is  evident  that  the 
ixnnts  in  contact  are  moving  with  the  same  velocity.  No 
hody  in  nature  is  perfectly  inrhtxiic  :  ami  hence,  at  the 
instant  of  greatest  compression,  the  I'liisflc  fiirri:i  of  resfi- 
fiifinn  are  l)rought  into  action  ;  and  during  the  second  part 
of  the  impact  the  mutual  pressure,  ])roduce(l  hy  the  ■Oa^^tic 
I'oriH'S,  which  were  hrought  into  action    iiy  the  compression 


y 


KLASTimV    OF  JIOVIES. 


373 


111  wliicli  tlio 
luce  an  cttVct 
an  iniinilt^ivt' 
[It  gi'iRTatc  a 
oi'  a  f^i-cond. 
1(1  siiMU'ratu  a 
III  tlio  I'iloets 
■d. 

en  two  l)0(lic'8 
jc't'oiv  ini})at't 
eoniiniin  tan- 
(I  the  line  (tf 
nintl.  Wlicn 
is  said  lo  !)(> 

)(in  tlio  other, 
taiit,  niiist  he 
ilie  third  law 
e  two  iiodies. 
al  cliaiigos  of 

af  two  jiarts  ; 
to  closer  I'oii- 
e  jiartieles  in 
fj  a  conijires- 
I creases  till  it 
I'lictinns.  thus 
r  c'<)iii|irossion 
lent  that  the 
velocity.  No 
hence,  at  tlie 
nrcfi  of  renti- 
le  second  part 
hy  the  .'lactic 
e  compression 


lUiring  the  tirst  part  of   the   impact,  tend   to  separate  the 
two  bo<lies,  and  to  restore  them  to  their  original  form. 

205.  Elasticity  of  Bodies.— Coefficient  of  Resti- 
tution.—It   aj. pears  from  experiment   that   hodies  may  lie 
eompressed   in   various  degrees,  and   recover  more  or  les, 
their  original  forms  after  the  coniiiressing  force  has  ceased, 
this  property  is   termed  daslicity.     The   fnrce  urging  the 
approaeli  of  bodies  is  called   the  fuvrr  of  comprcmoH  ;    the 
force   causing   the   hodies    to   separate  again   is  called  the 
force  of  irsfi/iifion.     Elastic  hodies  are  such  as  regain  a 
part  or  all  of  tlieir  original   form  when   the  compressing 
force  is  removed.     The  ratio  of  the  force  ..f  restitution  to 
that  of  comi.ressior.  is  called  the  Cuefficirm  of  h'rs/,/»/,o,i/- 
It  has  been   found   that    this  ratio,  in   the  same  bodies,  is 
constant  whatever  may  lie  their  velocities. 

When  this  ratio  is  unitv  the  two  forces  arc  e(iual,  and  (he 
body  is  said  to  ho  perfcrthj  vhixtic;  when  the  ratio  is  zero, 
or  the  force  of  restitution  is  m.thing,  the  body  is  saul  to  bb 
uoH-dasli<-;  when  the  ratio  is  greater  than  zero  and  less 
than  unitv.  the  bodv  is  said  to  be  mperfi"-ily  elastic.  There 
are  no  hodies  cither  perfectly  elastic  or  iierfeetly  non-clas- 
tic, all  being  more  or  less  elastic. 

In  the  cases  discussed  the  hodies  will  be  supi)osc<l  splier- 
ieal,  and  in  the  case  of  direct  impact  of  smooth  spheres  it 
is  evident   that  tliev  maybe  considered  as  particles,  since 

they  are  symmetrica'l  with  respect  to  the  line  joining  their 

The  theory  of  the  impact  of  hodies  is  chiefly  due  to 
Newton,  who"  found,  in  Ids  experiments,  that,  provided  the 
i,„,,.,ct  is  not  so  violent  as  to  make  any  sensible  iiideiitati.m 
i„  either  body,  the  relative  velocity  of  separation  after  the 
i,„paet  bears  a  ratio  to  the  relative  selocity  of  approach 
before   the   impact,   which   is  constant   for   the   same    two 


»  8imii:liiii('> 


iSllrd  CmMl'u'U'iil  uf  l^la-licily. 


Tixlliiiiiicr'BMi'ih..  p.  878. 


mm 


374 


DIRECT  IMI'AIT   <lF  r.\KI,ASTIc  liODIES, 


bodios.  In  Newton's  oxporiiui'iits,  liowevor,  the  two  bodies 
seem  always  to  have  been  I'ornied  of  tlie  same  sub- 
stance. He  fdiind  tbiit  the  value  nf  tiiis  ratio  (the  cof///- 
vicnt  of  ri'sfitulion),  fur  balls  of  compressed  wool  was  about 
J,  steel  about  tlie  same,  cork  a  little  less,,  ivory  |.  glass  \l. 
The  results  of  more  recent  experiments,  made  by  Mr. 
Ilodgkinson,  and  recorded  in  the  Hrpoit  of  tliv  BrUish 
Associiifio)!  for  ISSJf  show  that  tlie  theory  may  Iw;  received 
as  satisfactory,  with  the  exception  that  the  value  of  the 
ratio,  instead  of  being  (piite  constant,  diniinishes  when  the 
velocities  are  very  large. 

206.  Direct  Impact  of  Inelastic  Bodies. — .1  sphere 
of  ma.-<s  M,  movimj  with  a  vrlori/i/  v.  overtakes  and  impinges 
dirertty  on  another  sphere  of  iiiaxs  M',  niorin;/  in  the  same 
direrlicn  with  velocity  v',  and  at  the  instant  of  greatest, 
mutual  compression  the  spheres  arc  moving  with  a  common 
velocity  V.  Determine  the  motion  after  impact,  and  the 
impulse  during  the  compresswn. 

Ix't  R  denote  the  inipiilso  during  the  compression,  which 
acts  on  each  body  in  opposite  directions  ;  and  let  us  siip- 
])ose  \\w  bodies  to  be  moving  from  left  to  right.  Then, 
since  the  impulse  is  measured  by  the  amount  of  momentum 
gained  by  one  of  the  impinging  liodies  or  lost  by  the  other 
(Art.  202),  we  have 


Momentum  lost  by  M  =  M  {v  —  V)  =  L\ 
"       gained  by  J/'  ^  J/'  (  V-  v')  =  A', 
.-.     .J/(r-  V)  ^  .]f'(V-v'). 
Solving  (.'})  for  Two  get 


r 


wiiich  in  (1)  or  (2)  gives 


Afc  +  ^t',>' 
^f+^{''' 


(1) 

(4) 


le  two  bodies 
'  same  siih- 
;io  (the  cocffi- 
)ol  was  about 
y  i  glass  H- 
liuk'  by  Mr. 
f  f/ii'  iinfish 
y  111!  received 
value  of  the 
les  when  the 


s. — .1  sp/iere 
and  im.pi)tges 
in  tin;  .sdiiic 
t  of  (jrentrxt 
th  a  cniinnnn 
}act,   and  lite 

cssioii.  which 

let  IIS  siip- 

ight.     Then, 

if  nionientnni 

by  the  other 


Ji^  (1) 


(:5) 


(^) 


DIRECT  IMPACT  OF  LXKLASTIC  BODIES.  3T5 

(5) 


_  MM'  {i'  -  V) 
^-       M  +  M' 


liencff  the  common  vdonlieft  of  the  two  ladies  after  impact 
is  n/uat  to  t/ic  ahjfbraic  mm  of  ilteir  momenta,  divided  by 
the  snm  of  their  masses,  and  also,  from  (4),  the  whole 
momentum  after  impact  is  equal  to  the  sum  of  the  momenta 
before. 

Cor.  1.— Had  the  balls  been  moving  in  opposite  direc- 
tions, for  example  had  J/'  been  moving  from  right  to  left, 
(•'  would  have  been  negative,  in  which  case  we  would  have 

^^       Mv  -  M'v-  ,      ,,        MM'  {r  +  v')         ,  . 

From  the  first  of  these  it  follows  that  both  balls  will  be 

reduced  to  rest  if 

Mv  -  Mv; 

that  is,  if  before   impact   they   have   equal   and  opposite 
momenta. 

(j„j[,  >^._lf  M'  is  at  rest  before  impact,  v'  =.  0,  and  (-1) 

becomes 

Mv 


^  -  M  +  M'' 

If  the  masses  are  eciual  we  have  from  (4)  and  (0) 


r  = 


V  +  V 


or 


V  —  V' 


(7) 


(8) 


according  as  they  move  in  the  same  or  in  ojjposite  direc- 
tions. 

207.  Direct  Impact  of  Elastic  Bodies.— Win  n  ihc 

balls  are  clastic  the  problem  is  the  same,  up  to  the  instant 
of  tfreatest  compression,  as  if  they  were  inelastic;  but  at 


^ 


370 


DIRECT  IMl'ACT   OF  l.SKLASTIC  liODIES. 


this  instant,  the  force  of  restitution,  or  that  tendency  which 
elastic  bodies  have  to  regain  their  original  form,  begins  to 
throw  one  ball  forward  wit!"  the  same  nionientiim  that  it 
throws  the  other  back,  and  this  mutual  pressure  is  jn-opor- 
tional  to  R  (Art.  205). 

Let  0  be  the  coetlicient  of  restitution  ;  then  during  the 
second  part  of  the  impact,  an  impulse,  eU,  acts  on  eacii 
ball  in  the  same  direction  respectively  as  i^  acted  during 
the  com])ression.  Let  c,  and  i\  be  the  velocities  of  the 
balls  M  and  J/'  when  they  are  linally  separated.  Then  we 
liave,  as  before, 

Momentum  lost  by  M  z=  M{V—i\)  =  cR,       (1) 

gained  by  M'  =  M'  (r/  -  T')  =  cR.     (2) 

From  (1)  we  have 


i\  =  V  - 


eii 
M 


Mv  +  M'r'       _e.y' 


by  (4)  and  {'))  of  Art.  WG, 


M' 


JHTm'  <!  +  '')("  -"  )• 


Similarly  from  (2)  we  have 

M 


*'■' =  ^' +  irq:  J/' ^^  +0(^-"'); 


(J3) 


(4) 


v'hich  are  fhe  vclocifics  of  I  lie  //(tils  ir/icii  Jiunlhj  separated. 

These  results  may  be  more  easily  ol)tained  by  the  con- 
sideration that  the  whole  impulse  is  (1  -\- e)  R\  for  this 
liives  at  once  the  whole  iiiomeMtnm  lost  bv  iV  or  gained  bv 
.1/'  during  compression  and  restitution  as  follows: 


M{v-,',)  =  (1  +e)R, 


(5) 


DIES. 

Jiidciicy  w  liich 
orni,  iH'giiis  tu 
entiim  that  it 
lire  is  ])roi)or- 

011  during  the 
,  acts  oil  eacli 
'  ucted  during 
locities  of  the 
ed.     Then  we 


=  en,      (1) 

)  =  elt.     U) 


') 

■))  of  Art.  -zm, 


v'); 


(4) 


!/  separdled. 
d   by  tlie  con- 
)  li ;   lor  this 
or  gained  hy 

(5) 


VIIihCT  IMl'ACT   OF  ISELASTIV   BODIES.  3T7 

and  M' {v,' -  v)  ^  {I  +  e)  R.  (0) 

Substituting  in  ("))  and  (C)  the  value  of  R  from  (5)  of  Art. 
•^OG,  \vc  have  the  values  of  c  and  i\'  immediately. 

C'ou.  J. — If  the  l)alls  are  moving  in  opposite  directions, 
/•'  becomes  negative.  If  tlie  balls  are  non-elastic,  e  —  0, 
and  (3)  and  (4)  reduce  to  (4)  of  Art.  WG,  as  they  should. 

Cor.  -2. — If  the  balls  are  perfectly  elastic,  e  =  1,  and  (;3) 
and  (4)  become 

2M' 


i;  =  r 


V  + 


M  +  M 


{V  -  v'), 
-  {v  -  v'). 


(7) 


(«) 


M  4-  M 

CoK.  ;i. — Subtracting  (4)  fnmi  (:])  and  reducing,  we  get 

,.,_,.,'  ^  r-r  -(I  +  e){v-v'\ 

=  -c{c-r').  (9) 

Hence,  the  relative  velocity  after  impart  is  —  e  times  tlic 
relatire  relocity  before  impart. 

(.'oii.   4.  — >[ultiplyiiig  {i)  and  (4)  by  .)/ and  J/',  respect- 
ively, and  adding,  we  get 


Mr,  -f  M'r;  =  Mr  +  Mr'. 


(10) 


Hence,  as  in  Art.  •^()(i,  the  alyehrair  .sinii  nf  tlie  momenta 
after  impart  is  the  same  as  tiefire ;  i.e.,  there  is  no  mo- 
iiiriitiim  lost,  which  of  course  is  a  direct  consequenoe  of  the 
tliird  law  of  motion  (Art.  Kii)). 

Coii.  .'i. — Suppose  (•'  —  0,  .so  that  the  body  of  mass  .)/, 
moving  with  velocity  /',  impinges  on  a  body  of  mass  .1/'  at 
rest,  then  (3)  and  (4)  become 


ito 


378 


LOSS   OF  lilNKTlC  K.\ERGY. 


_  M  -  eM' 


una 


Hence  the  body  which  is  struck  goes  (iiiwards ;  and  tlie 
Htnking  body  goes  onwards,  or  stops,  or  goes  backwards, 
according  as  J/ is  greater  than,  eijual  to,  or  less  than  eM', 
If  J/'  =  eM,  then  (11)  becomes 


r,  =  (1  —  (')  V,    and     r,'  =  v. 


(1^) 


CoK.  C— If  M  =  M  and  r  =  1  ;  that  is,  if  the  l)alis 
are  of  equal  mass  and  perfectly  elastic,*  then  (T)  and  (8) 
become,  respectively, 


v',     and     r,' 


(13) 


that  is,  the  balls  interchange  their  velocities,  and  the 
motion  is  the  same  as  if  they  had  jjassed  through  one 
another  without  exerting  any  niutiud  action  whatever. 

Cor.  7. — If  M'  he  infinite,  and  v'  =  0,  we  have  the  case 
of  a  ball  impinging  directly  upon  a  Jixed  surface ;  substi- 
tuting these  values  in  (3)  it  becomes 


V,  =  —  ev] 


(14) 


that  is,  the  ball  rebounds  from  the  fixed  surface  with  a  veloc- 
ity e  times  that  with  which  it  impinyed. 

208    Lobs  of  Kinetic  Energy  f   in  the  Impact  of 

Bodies.— Scpiaring  (it)  of  Art.  ^(»7,  and  multiplying  it  by 
MM' ,  we  iiave 

MM'  (r,  -  lu'Y  =  MM'  c'  (/-  -  v'f 

-  MM'  {r  -  v)i  -  (1  -  ^^)  MM'  {r  -  v'y.        (1 ) 


'  TliU   is   llic   usual   iihrascolojiy,  hut   iiiiKU'ttdiiif;,  Eiicy,  Brit.,  Vol.  XV,  All. 
Mi'cUV. 

t  See  Art.  189. 


■D 


M' 


(11) 


irds ;  mid  tlu' 
ics  backward.^, 
e^s  than  eM'. 


(1^) 

is,  if  the  l)all« 
n  (T)  and  (8) 


(13) 

hies,   and    tlii' 
through  one 
iliatcver. 

havo  the  case 
rfuce ;  siibsti- 


(U) 
c  with  a  veloc- 

i  Impact  of 

Itiplying  it  by 

f 

-vy.      (1) 

:iil.,  Vi)l.  XV.  All. 


LOSS    OF  KISETIC  ESEHGY.  :j79 

Squaring  (10)  of  Art.  :^0T,  wo  have 

(J/f,  +  M'v^f  =  {Mv  +  M'v'f.  (3) 

Adding  (1)  and  (•■.*),  we  get 

{M  +  M)  (J/r,2  +  J/(','2)  =.  {M  +  M')  (Mv^  +  M'v'^) 

-  (1  -  e^)  MM'  {v  -  vy  ; 

-i{l-'''^)-j^^77(^-i'T,(3) 

the  last  term  of  which  is  the  loss  of  kinetic  energy  by 
inii)act,  since  c  can  never  be  greater  than  unity.  Hence, 
there  is  always  a  l()s.s  of  kinetic  energy  by  impact,  exee})t 
when  c  —  1,  in  wliich  ca.se  the  loss  is  zero;  L  e.,  when  the 
coefficient  of  restitution  is  unity,  no  kinetic  energy  is  lost. 
When  c  =  0  the  loss  is  the  greatest,  and  equal  to 


i 


MM' 


M  +  M 


W  ^^  -  ^')'- 


(4) 


From  (3)  we  see  that  during  compression  kinetic  energy 

to  tiie   amount  of   i   il^y'  (''  —  '')'  ''^  ^^^^ '    '^"*^  ^'^''" 
during  restitution,  c'  times  this  amount  is  regained. 

Kkm.— From  the  theory  of  kinetic  energy  it  ai)i)ears 
that,  in  every  case  in  which  energy  is  lost  by  resistance, 
heal  is  generated  ;  and  from  Joule's*  investigations  wo 
learn  that  the  (jiumtity  of  heat  so  generated  is  a  lu'rfeetly 
delinite  rt/uiralrtit   for   the  energy   lost;  and  also  tiiat,  in 

♦  Sic  "The  Corri'liitioii  (lint  ruii-crvnlidii  of  Forres,"  liv  nclnilinllz,  Fur.iduy, 
Melil;;,  clc.  ;  a\m  "  Heal  ii^  ii  Mode  cif  Motion."  liy  I'rof,  Tvndull  Also  ».  SIcwiirf.i 
"  Conservation  of  Eiier{;y." 


mm 


380 


OBLIQIE   IMI'ACT. 


any  natural  action,  there  is  never  a  development  of  energv 
whicii  cannot  be  accounted  for  by  the  disappearance  of  an 
equal  amount  elsewhoie  by  mean.<  of  some  known  phvsical 
atrency.  Hence,  the  kinetic  energy  which  api)ears  to  be 
lost  in  the  above  cases  of  impact,  is  only  transformed, 
partly  into  heating  the  bodies  and  the  surrounding  air,  and 
partly  into  sonorous  vibrations,  as  in  the  impact  of  a  ham- 
mer on  a  bell. 

209.  Oblique  Impact  of  Bodies.— The  only  other 
case  which  we  shall  treat  of  is  that  of  oblicjue  impact  when 
the  bodies  are  spherical  and  pei'feetly  smooth. 

A  particle  iinpnir/cs  iri/Ii  a  fjircii  vehcify,  and  in  a  (jiven 
direction,  on  a  smooth  plane;  required  to  determine  the 
motion  after  impact. 

Let  AC  represent  the  direc- 
tion of  the  velocity  before  im- 
])aet,  meeting  the  plane  at  C, 

and    CB    the    direction   after 

impact.      Draw    CD   perpen- 
dicular   to   the   plane  ;    then 

since  the  plane  is  smooth  its  impulsive  reaction  will  be 
along  CD. 

Let  V  ar.d  i\  denote  tbe  velocities  before  and  after 
impact,  respectivelv  :  and  let  a  and  (i  denote  the  angles 
ACD  and  IK'D. 

Resolve  v  along  the  plane  and  perpendicular  to  it.  The 
former  will  not  be  altered,  since  the  impulsive  force  acts 
l)erpendi{Milar  to  the  ])lane ;  the  latter  may  be  treated  as  in 
the  case  of  direct  imjiact,  and  will  therefore,  after  impiict. 


Fig.87 


be  e  times   what   it  was  before  (Art.  20;,  Cor.  T) 

resolving    ^>,    alon 

have 


Hen 


and    perpendicular    to   the   plane,   we 
r,  sin  /i  —  V  sin  «,  (1) 

(2) 


'c,  cos  fi  =z  —  e  'V  cos  «. 


I'lit  of  I'liergv 
araiitr  of  an 
)\vii   ])Iiysiciil 

ippears  to  lie 
traiisfonncd, 
ding  air,  and 

act  of  a  iium- 


0   only   otiicr 
imj)a(;t  wlien 

1(1  ill  a  f/iroi 
determine  the 


py 


1.87 

lotion  will  be 

re  and    after 
te  the  anglo.s 

!•  to  it.  'I'lic 
ivo  forci'  acts 
treated  as  in 
after  impMct. 
.  T).  Hence, 
le    plane,    we 

(1) 


OBLKiVE  JMl'AVT. 

Dividing  ('-i)  by  (1),  we  got 

cot  ^}  =  —  c  cot  «. 
S(piaring  (1)  and  {'I),  and  adding,  we  get 

i\^  =■  ('2  (si  11^  a  +  f2  cos^  «). 


381 


Ci) 


(^) 


Tims  (3)  determines  tlie  diredioiu  and   (4)  tiie  mfif/iiitndc 
of  the  velocity  after  impact. 

The  angle  ACl)  is  called  the  (iii(/le  af  incidence,  and  the 
angU'  BCD  the  angle  of  reflexion. 

CoH.  1. — If  the  elasticity  be  perfect,  or  ';  =  1,  we  have 
from  (3)  and  (4), 


cot  /3  =  —  cot  «,  or  /J  = 


and 


("',  or  r, 


(5) 
CO 


Hence,  in  perfertlii  elastic  hulh  the  (inf/Ie.'^  of  incidence 
and  reflexion  are  nmnrrienlli/  ci/iiid.  and  tite  velocities  bi'forv 
and  after  impact  are  rquaJ.  This  is  the  ordinary  rule  in 
the  case  of  a  billiard  ball  striking  the  cnsiiion. 

Coll.  X'.— 8ui)pose  r  =  0;  then  from  (3),  H  ^  'Mf. 
Tims,  if  there  is  no  elasticity,  the  body  after  impact  moves 
along  the  plane  with  the  velocity  v  sin  «(. 

If  a  =  0,  so  that  the  impact  is  direct,  we  have  from  (4), 
r,  =  ev  ;  i,  e.,  after  the  impact  the  body  roounded  along 
its  former  course  with  e  times  its  former  velocity. 

If  fc  =  0,  and  e  =  Oj  then  from  (4),  r,  =  0,  and  the 
body  is  l)ronght  to  rest  by  the  imj)act. 

Sen. — Of  course  the  results  of  this  iirticleare  applieal)le 
to  cases  of  impact  on  any  smooth  surface,  by  substituting 
for  tlie  plane  on   which  the  impact   has  been  supposed  to 


ns-^ 


Olil.KilK   JMI'Acr   iiF   711  o    SMOOTH  Sl'llHUES. 


tiiki'  pliR'L'  tlio  jdane'  which  is  tungoiit  to  the  surfi'.cc  at  the 
j)()int  of  iinpacl. 

210.  Oblique  Impact  of  Two  Smooth  Spheres. — 

Ticti  siniiDlh  sjilicn-s.  murimj  in  ijiven  dircrtioiix  iiuil  irilli 
ijiri-n  rcloriUrs,  impinge;  to  deteDiiine  the  impulse  and  the 
•ntbuequent  motion. 

Q 

Let  the  masses 
of  the  sj)lu'res  be 
AL  M'  ;  their  cen- 
tres (',  C;  tiieir 
velocities  before 
impact  V  and  /•', 
and    after    inij)act 


Fig.  88 


V,  and  i\  .  Let  ED  lie  the  line  wiiicli  joins  tlieir  centres  at 
the  instant  of  impai't  (called  the  line  of  impact):  C'A  and 
C'H  the  directions  of  motion  of  the  imjjinjrinij  si)iiere,  M, 
before  and  after  impact  ;  and  C'A'  and  C'B'  those  of  the 
other  spiiere:  let  «,  a'  be  the  anjiflcs,  ACD  and  A'C'D, 
which  the  ori<^inal  directions  of  motion  make  with  the  line 
of  impact:  /i  i3'  the  angles.  BCD  and  B'C'D,  which  their 
directions  make  after  the  impaci. 

It  is  evident  that,  since  tiie  spheres  are  smooth,  the 
entire  mntiial  impulsive  pressure  takes  ])lace  in  the  line 
joininir  the  centres  at  the  instant  of  impact.  Ijct  7^  be  ti-,e 
imi)ulse.  and  e  tiie  coeflicieni  of  restitution.  Resolve  all 
the  velocities  alon<i  the  line  of  imi)act  and  at  right  angles 
to  it  ;  the  latter  will  not  be  affected  by  the  inii)act.  and  the 
former  will  l)e  affected  exactly  in  the  same  way  as  if  tlie 
impact  had  l)een  direct.  Hence,  since  the  velocities  in  the 
line  of  impact  are  r  cos  «,  r'  cos  «',  ?',  cos  /3,  ;','  cos  /3',  we 
iiave.  by  substituting  in  (:5)  and  (4)  of  Art.  '-iUT, 


r,  cos  /i  =  r  cos  « 


.1/' 
M+  M 


-,  (1  +1')  {''  t'os  « —  /•'  cos  «'),  (1) 


lERES. 
irfi'.cc  at  the 


Spheres. — 

II -t  mill  irilli 
Hilse  uvd  tin: 


KXAMI'hKS. 


^ft' 


'ir  centres  at 
ft):  CA  and 
(T  splierc,  M, 
t-hoso  of  the 
and  A'Cl), 
ivitli  tlie  lino 
,  wliich  their 

smooth,   the 

in  the   line 

Lot  li  be  ti-e 

Resolve  all 

riglit  an<:le.s 

)acf,  and  the 

way  as  if  tlie 

)cities  in  tlie 

i\  cos  /3',  we 


•'ens  «'),(!) 


3s:{ 
) 


,•/  eos/i'  =:::  c' COS  «' +   ^f',     ]/'(•+'')  ('' ^'''S  «- '''  ^OS  «'),    {'i , 

irhich  lire  lliv  fiiKtl  rrlocilii'.s  of  I  In'  two  spliercs  aluinj  llir  h'lii' 
of  iin/iaci  ED. 

Also,  from  (."))  of  Art.  '-iOii,  we  obtain  by  substitntion. 

1/  1/' 


E 


MM'     ,  ,. 

-^— — ^,  (<•  cos  «  —   V    COS  «  ), 
Ji    +   M 


(5) 


Jl     -t-     JH 

(See  Tait  and  Steele's  Dynamics  (jf  a  Particle,  p.  323.) 

Cor.  1.— Multiplying  (I)  by  .)/,  and  (-2)  by  M',  and  add- 
ing we  get 

Mi\  cos  li  +  M'i\  cos  fl'  =  Mn  cos  «  +  M'v  '  cos  «',  (4) 

wiiich    shows   that    Ilir   moineitlum    af  the  .si/slein  nsulred 
aJiniij  the  line  of  impact  is  the  smnv  after  impact  as  before. 

Cou.  -2.— Subtracting  Ci)  from  (1)  we  ()l)tain, 

r,  cos  /3  —  r,'  cos  /J'  ■=  —  e  {o  cos  u  —  v'  cos  «').     (5^ 

That  is,  the  relative   rehcili/.  resotred  alomj  the  line  o, 
impact,  after  impact  is  —  e  times  its  calue  before. 

EXAMPLES. 

1.  A  body*  weigiiing  3  lbs.  moving  with  a  velocity  of 
10  ft,  per  second,  impinges  on  a  body  weighing  l  lbs.,  and 
moving  with  a  velocity  of  3  ft.  per  second  ;  find  the  com- 
mon velocity  after  im|)act.  Ans.  Tl  ft.  per  second. 

'.'.  A  body  weighing  7  lbs.  moving  11  ft.  per  secnni. 
imi)inges  on  another  al  rc<t  weighing  l."i  Uts.;  lind  the  com- 
mon velocity  after  imjiact.  A)is.   31  ft.  per  second. 

*  The  bodies  Brc  iIlt■la^-tic  iiiili's-  (illieiwi-f  st.ntod.  The  first  27  f.\aiui)l.';4  mv  in 
direct  Impact. 


m 


38-1 


IX.l.UrLKS. 


:].  \  body  weigliiiif,'  -I  Hi.s.  moving  1)  ft.  per  a'foinl. 
iiiipinges  on  another  body  weigliiug  2  lbs.  and  moving  in 
the  op])osite  direction  with  a  velocity  of  .j  ft.  per  second; 
find  tlie  common  vclticiiy  after  impact. 

A 11^.   4^  ft.  per  second. 

4.  .\  body,  M',  weighing  5  lbs.  moving  7  ft.  per  second, 
is  imi)inged  njion  by  a  body,  .1/,  weighing  0  lb.«.  and  mov- 
ing in  the  same  direction  :  after  impact  the  velocity  of  J/' 
is  doiil)led:  iind  the  velocity  of  .]/  before  impact. 

Ans.    U)f.  ft.  per  second. 

5.  Two  bodies,  weighing  2  lbs.,  and  4  lbs.,  and  moving  in 
the  same  direition  with  the  velocities  of  0  and  !1  ft.  respec- 
tively, impinge  iijton  each  other  ;  find  their  common 
velocity  after  im|)act.  -!//•'>.   8  ft.  per  second. 

().  X  weight  of  i  lbs.,  moving  with  a  velocity  of  "^O  ft. 
per  second,  overtakes  one  of  ')  lbs.,  moving  with  a  velocity 
of  0  ft.  per  second  ;  find  the  common  velocity -after  impact. 

Alls.  !l^  ft.  per  second. 

T.  If  the  same  bodies  mcf  with  the  same  velocities  find 
the  common  velocity  after  impact. 

,l//.s'.   •i\  ft.  i)er  second  in  the  direction  of  the  first. 

8.  Two  bodies  of  dilTerent  mas-jes,  are  moving  towards 
each  other,  with  velocities  of  In  ft.  and  \'i  ft.  per  second 
respectively,  and  contiinie  to  move  after  impact  with  a 
velocity  of  1  •  3  ft.  per  second  in  the  direction  of  the  greater; 
c.npare  their  masses.  Ans.  As  ;J  to  •^. 

1).  A  bodv  impinges  on  anotiier  of  twice  its  mass  at  rest; 
show  that  the  itupinging  body  loses  two-thirds  of  its 
velocity  by  the  iin[iact. 

1(1.  Two  bodies  of  uneijnal  masses  moviiig  in  opposite 
directions  with  momenta  iinmerically  ecpial  meet  ;  show 
til, it  the  momenta  are  numerically  eijual  after  imjiact. 


^ 


t.  i)er  fici'oiul. 
11(1  moviiif,'  in 
t.  per  second  ; 

per  seeonil. 

n.  per  second, 

i  ll).s.  and  niov- 

vclocity  of  J/' 

act. 

.  per  .■second. 

and  niovin<;  in 
x\  0  ft.  re.*  pee - 
heir    common 
per  second. 

■locity  of  W  ft. 
"itii  a  velocity 
y  after  impact. 
,  i)er  second. 

velocities  find 

of  the  tirst. 

loviiif^  towards 
ft.  per  second 
imjiact  with  a 
of  the  greater; 
s.  As;Jto'^. 

s  mass  at  rest; 
-thirds    of    its 


)i<r  in  opposite 
1  meet  ;  sIkjw 
r  imjiact. 


^ 


EXAMri.KS. 


:5K.') 


n.  A  hody.  .1/,  wei.iriunj:  10  ihs.  moving  8  ft.  per  second, 
impinges  on  M\  weighing  <1  His.  and  moving  in  the  same 
direction  .■■>  ft.  per  second  ;  tinil  their  velocities  after  impact, 
snpposing  (=1. 

Ans.  Velocity  of  .1/  =  5J  ;  velocity  of  .)/ '  =  Sj|. 

Vi.  A  liody.  M.  weighing  4  lbs.  moving  0  ft.  per  second, 
meets  M  weighing  8  lbs.  and  moving  4  ft.  jier  .second; 
find  their  velocities  after  imi)act,  v  —  1. 

An<.  Each  body  is  retlected  back,  J/ with  a  velocitv  of 
7^  and.l/'  with  a  velocity  of  -11 

13.  Two  balls,  of  4  and  H  lbs.  weight,  impinge  (m  each 
other  when  moving  in  the  same  direction  with  velocities  of 
0  and  10  ft.  re.<])Oclively  ;  find  their  velocities  after  impact, 
snpposing  e  -  |.  ^l''--   10-<>8  and  !)-;i8. 

14.  Find  the  kinetic  energy  lost  by  imjiact  in  examide  o. 


Alls. 


h\- 


15.  Two  bodies  weighing  40  and  dO  lbs.  and  moving  in 
the  same  direction  with  velocities  of  IC  and  '^t;  ft.  resi)ee- 
tively,  imjiinge  on  each  other,  liiul  the  loss  of  kinetic 
energy  by 'in jiact.  ^l"''-  -ST-o. 

IG.  An  arrow  shot  from  a  bow  starts  oil  with  a  velocity 
of  120  ft.  per  second;  with  what  velocity  will  an  arrow 
twice  as  heavy  leave  the  bow.  if  sent  off  with  three  times 
the  force?  ''"*'•   180  ft.  per  second. 

17.  Two  balls,  weighing  8  ozs.  and  (i  ozs.  respectively, 
are  simultaneously  projected  upwards,  the  former  rises  to  a 
height  of  ;i'24  ft.  and  the  latter  to  '-ioG  ft.:  compare  the 
forces  of  projection.  ♦'«•''■•   As  3  to  2. 

18.  A  freight  train,  weigliing  200  tons,  and  traveling  20 
miles  i)cr  hr.  runs  into  a  i)asscnger  train  of  50  tons,  stand- 
ing on  the  same  track;  find  the  velocity  at  which  the 
remains  of  tlie  passenger  train  will  be  propelled  along  the 
track,  supposing  c  =  \.  Ans.   19-2  miles  per  hr. 


iftM 


380 


EXAMPLES. 


10.  Tliere  is  a  row  of  ton  jiorfectly  elastic  belies  whose 
masses  increase  geoinetncally  l)y  the  constant  ratio  3,  and 
the  first  impinges  on  the  second  witii  the  velocity  of 
0  ft.  i)C'r  second  ;  tind  tiic  velocity  of  the  last  Imdy. 

Alls,    sf  J  ft.  i)er  second. 

20.  A  body  weighing  .5  lbs.  moving  with  a  velocity  of  1-f 
ft.  per  second,  hnpinges  on  a  body  weighing  3  lbs.,  and 
moving  with  a  velocity  of  8  ft.  per  second;  lind  the  veloci- 
ties after  impact  supposing  p  =  {.  Aiix.   11  and  13. 

'i\.  Two  bodies  are  moving  in  the  same  direction  with 
the  velocities  7  aiu.  h  ;  and  after  impact  their  velocities 
are  5  and  G;  lind  c,  and  the  ratio  of  their  masses. 

J  HA',  c  =  \\  M'  =  'iM. 

'^i.  A  l)ody  weighing  two  lbs.  impinges  on  a  body  weighing 
one  lb.;  c  is  i,  show  that  r,  =  \{r  +  r'),  and  that  r,'  =  v. 

To.  Two  bodies  moving  with  numerically  equal  velocities 
in  o|)posite  directions,  impinge  on  each  other;  the  result  is 
that  one  of  tliem  turns  back  with  its  original  velocity,  and 
the  other  follows  it  with  half  that  velocity;  show  that  one 
body  is  four  times  as  heavy  as  the  other,  and  that  c  =  J, 

24.  A  strikes  H,  which  is  at  rest,  and  after  imj>act  the 
velocities  are  numerically  ctiual;  if  r  be  the  ratio  of  B's 

mass  to  A's  mass,  show  that  e  is      "  ,  ,  and  that  B's  mass 


1 


is  at  least  three  times  A's  mass. 


25.  A  body  impinges  on  an  eipial  body  at  rest;  show 
thill  the  kinetic  eiu'rgy  before  impact  cannot  be  greater 
than  twice  the  kinetic  I'liergy  after  impiict. 

2(!.  A  series  of  perfectly  elasiic  balls  are  arranged  in  the 
Slime  straight  line;  oiu'  of  Iheiii  impinges  on  the  next, 
then  this  on  the  n^'xt  iiinl  so  on;  show  thiit  if  (heir  musses 
form  a  geomelrie  progression  of  which  the  common  ratio 


jo'lies  whose 
ratio  3,  and 

I  velocity  of 
jdy. 

)or  second. 

L'iocity  of  14 
<;  3  lbs.,  and 
d  the  vcloei- 

II  and  13. 

rection  with 

eir  velocities 

ei*. 

M'  =  2M. 

xly  wcigiiing 
that  r,'  =  V. 

ijal  velocities 
the  result  is 
velocity,  and 
II I w  that  one 
lat  c  =  J. 

'r  imj)act  tlie 
ratio  of  B's 

hat  B's  mass 


t  rest ;  show 
it  l)e  greater 

angcil  in  tlie 
i)n  tlie  next, 
I  heir  musses 
Diunion  ratio 


KX  AMI'LES. 


387 


.<ion  nl'  wiiieii  the  common  ratio  is 


is  /•.  tlieir  velocities  after  imiiact  form  a  geometric  progres- 

2 
'•'+  l" 

■jr.  A  l)iiil  falls  from  rest  at  a  height  of  20  ft.  above  a 
fixed  horizontal  plane:  find  the  height  to  which  it  will 
rebound,  c  being  J.  and  //  being  '-Vl.  Ans.  l\\  feet. 

28.  A  ball  impinges  on  an  e(iual  hall  at  rest,  tlie  elas- 
ticity being  perfect;  if  the  original  direetion  of  the  strik- 
ing ball  is  inclined  at  an  angle  of  4.V'  to  the  straiglit  line 
joining  the  centres,  determine  the  angle  between  the 
directions  of  motion  of  the  striking  ball  before  and  after 
impact.  Ann.  45". 

29.  A  ball  falls  from  a  height  //  on  a  horizontal  jdano, 
and  then  rebounds;  find  the  height  to  which  it  rises  in  its 
ascent.  •'"*'•  f^/'-. 

30.  A  ball  of  mass  .lA,  inqiinges  on  a  ball  of  mass  M'.  at 
rest  ;  show  that  the  tangent  of  the  angle  between  the  old 
and  new  directions  of  the  motion  of  the  imping; 'g  body  is 

1  4-  (!  M'  sin  2fe  _ 

~Y    '  M  +lr  (sin-  H  —  r  eos^  «)' 

31.  A  ball  of  mass  i¥im])inges  on  a  liall  of  nuiss  M'  at 
rest  ;  find  the  comlition  in  order  that  the  directions  of 
motion  of  tiie  impinging  ball  iH'fore  and  after  impact  may 


be  at  right  angles. 


A  UK.  tan^  «  = 


Tr  +  M 


32.  A  ball  impinges  on  an  eiiual  ball  at  rest,  tlu' angle 
between  the  old  and  lu'w  directions  of  motion  of  the 
imiiin'Mnsr  ball  is  <10  ;  find  the  velocity  after  imitact.  c 
i.('in<''  1.  Aii>.   r  sin  :iO". 

3;;.  A  hall  imiiinges  on  an  e<iual  hall  at  rest,  >■  being  1  ; 
lind  the  condition  under  which  the  velocities  will  l)e  e(|ual 
after  impact.  Ans.  «  =  -iS" 


388 


KXAMfhES. 


34.  A  hall  is  projectod  I'roiii  I  lie  middle  jioiiit  of  one  sidy 
of  a  billiard  table,  so  as  to  sirilu'  tirst  an  adjuet'iit  side,  and 
tlu'ii  till'  middle  puint  of  the  sidi'  o|)posite  to  tliat  from 
wliieM  it  started:  iiiid  where  the  hail  must  hit  the  adjaeeiit 
side,  its  lenglh  heinfi^  b. 

Alls.   At  the  distance  from  the   end  nearest   the 

J  4-  <' 

ojiposite  side. 


iiit  of  one  sidy 
ict'ut  silk',  ami 
'  to  tliiif  from 
it  the  inljiic'i'iit 

1(1  iicurt'sl   the 


CHAPTER     V. 

WORK    AND    ENERGY. 

211.  Definition  and  Measure  of  Work.— Ho/ ^  / 
flic  jirodHdiiiH  of  iii(i/i(in  <i(/(iiiist  ri'si.stdiirr.  A  force  is  said 
to  lid  iri,rl\  if  it  moves  tiie  body  to  which  it  is  ai)i»lied  : 
and  tlie  woik  done  by  it  is  measured  by  the  product  of  the 
force  into  the  si»aee  tlirouj,di  wliieh  it  moves  the  Itody 
(Art.  101.  l{eni.). 

Tims,  the  work  done  in  liflinj,' a  weigiit  ti\rimgli  a  ver- 
tical distance  is  proportional  to  the  weight  lifted  and 
tiie  vertical  distance  tlirongh  whicli  it  is  lifted.  Thr  unit 
i,f  tntik-  used  in  Kiiglaiid  and  in  this  country  /n  Unit  which 
i.<  ri'ijuiri'il  Id  oirironii'  Ihr  irrii/hf  of  a  pinnnl  Ihrniiijli  Ihr 
rrrliriil  lin(jht  (»/"  ^«  /'/w/,  and  is  called  r^ /(w/-7'""«^/.  For 
instance,  if  a  weight  of  10  lbs.  is  raised  to  a  height  o'' 
T)  ft.,  or  T)  lbs.  raised  to  a  height  of  10  ft..  .V)  fool-pounds  o- 
work  must  have  been  e\i)cnded  in  overcoming  the  resist- 
ance of  gravity.  Similarly,  if  it  re(piiros  a  force  of  hi)  lbs. 
to  move  a  load  on  a  horizontal  plane  over  a  distance  of 
100  ft.,  5000  foot-pounds  of  work  must  have  been  done. 
If  a  carpenter  urges  forward  a  |)lane  through  3  ft.  with  a 
force  of  VI  ll)s..  he  does  'M\  foot-pounds  of  work  ;  or.  if  a 
weight  of  1  ll's.  desci'uds  tlirongb  10  fl.,  gravity  does 
TO  foot-pounds  of  work  on  il. 

Hence,  the  nnmiier  of  units  of  work,  or  foot-pounds, 
necessary  to  overcome  a  lonsMani  resistance  of  /'  iiounds 
through  a  distance  of  ,S'  feet  is  e(pial  to  the  jirotlitct  /'>'. 

From  this  it  appears  that,  if  tlu'  point  of  application 
move  always  pcr])endicular  to  the  direction  in  which  the 
force  acts,  siu'h  a  force  does  no  work.  Thus,  no  work  in 
dot;,'    by   gravity    in    the    case  of   ii   ))article  moving  on  u 


:U)0 


WoiiK  iinxh:  iiy  a  foiice. 


liorizDiital  piano,  and  wlicn  a  i)artielo  movos  on  any  smooth 
^nl•^ace  no  work  is  dono  hy  tliu  force  wliicli  llic  surface 
exerts  npon  i(. 

Neither /o/Y'c  nor  iiuiUdh  iihnie  is  sulTieient  to  constitute 
iroik  ;  so  tiiat  a  man  wlio  nieri'ly  snpiMjrts  a  h);id  without 
moving,'  it,  does  no  work,  in  the  sense  in  wliicl)  thai  term  is 
used  mei'hanieally,  any  more  tliiin  a  cohimn  does  whicii 
sustains  a  heavy  weight  upon  its  summit. 

If  a  body  is  moved  in  the  direction  oppnxile  to  that  in 
wiiich  its  weight  acts,  the  agent  raising  it  (h)es  work  upon 
it,  wliile  tlie  work  dtjue  l)y  tiie  eartii's  attraction  is  ncud- 
tirp.  \Vhcn  the  work  (h)ne  by  a  force  is  negative,  i.  c, 
when  the  point  of  applii'ation  moves  in  the  direction  oppo- 
site to  that  in  which  the  Ibrce  acts,  this  is  fre<picntly 
expressed  by  saying  tluit  work  is  done  (if/(ti)isf  tlie  force. 
In  tiie  above  case  work  is  done  by  tlie  force  lifting  the 
liody,  and  (Kjit'uisl  the  earth's  attraction. 

212.  General  Case  of  Work  done  by  a  Force.— 

Wiien  eitlier  the  magnitude  or  direction  of  a  ibrce  varies,  or 
if  lioth  of  them  vary,  the  work  done  by  the  force  during  any 
finite  disjtiacement  cannot  be  detlned  as  in  Art.  211.  Iii 
this  case  tlie  work  done  during  any  indelinitely  small  dis- 
))lacemenl  may  be  found  liy  s\ip})osing  tiie  magnitude  and 
direction  of  the  force  constani  during  llie  displacement,  and 
linding  tlie  work  done  as  in  Art.  "ZW  ;  then  taking  the  sum 
of  all  such  elements  of  work  done  during  the  consecutive 
small  displacements,  which  together  make  up  the  finite 
displaeenient,  we  ol)taiii  the  wiioji'  work  done  by  the  force 
during  siu'h  liuite  displacemeni . 

Thus  let  a  forcp,  P  act  nt  a  point,  0,  in  tlip  dirertioii  OP  {Fij;.  50), 
1111(1  let  us  suiiposo  the  point,  (),  to  move  into  iiiiy  otlier  position,  .1, 
very  ncnr  0.  If  0  lie  tlio  tingle  lictwccn  the  direction,  OP,  o(  the 
force  iind  the  direction,  OA,  of  the  diHjtluceiuent  of  the  |)()int  of  appii- 
ciition,  then  the  jtruduct,  /*•  OA  cos  0,  is  culled  the  work  done  by  the 
force.  If  we  dro))  n  perpcndiculiir,  AN,  on  OP,  the  work  done  liy  the 
force  1b  also  etjunl  to  tlu'  product  POiV,  where  ON  in  to  !»■  esti- 


11  any  smooth 
li   tlic  siirfiice 

to  coiistitiitc 
L  1(111(1  witlioii! 
Ii  lliiu  t(.'riii  iti 
11  do(.'S  wliirh 

He  to  that  in 
ic's  work  upon 
c'tion  is  ni'fia- 
lepativf,  i.  c, 
iroction  o])po- 
is  frc(juently 
)isf  tlio  force, 
•ce  lifting  the 

r  a  Force. — 

force  varies,  or 
I'ce  ihiring  any 

Art.  211.  ll. 
tely  small  ilis- 
lagnitiide  ami 
)laeenieiil.  and 
akiiig  the  sum 
le  consecutive 

up  the  finite 
)  by  the  force 

ion  OP  (FifT.  50), 
tlier  position,  .1, 
tioii,  OP,  ol  tlif 
ic  |H)int  of  a|)|)ii- 
■•ovk  (loiic  by  tile 
vorlt  done  liy  tlie 

)N    is    to    i)c  OHti- 


MKAsnn:  of  woiCK. 


31(1 


mated  ns  positive  whon  in  the  dircrtioii  of  tlio  loice.  If  scvoral  forct-s 
act,  tin'  worli  done  by  eacli  can  I'C  found  in  tlic  same  way  ;  and  tlio 
sum  of  all  tliesM^  i.s  the  work  done  l)y  tlic  \vliol(>  .sysK^in  of  forcos. 

It  appeals  from  tiiis  that  tlie  vvorlt  done  by  any  fonv  duiiiiK  an 
infinitesimiil  displacement  of  the  iioint  of  application,  is  the  iiroduct 
of  the  lesolvd  part  of  tlie  force  in  tlie  direction  of  tiie  displacement 
into  the  disphiceineiit  ;  and  tliis  is  tlie  same  as  tlie  cirtioi/  nmiuviit  (d' 
the  force,  which  has  been  described  in  Art.  101.  In  Statics  we  are 
concerned  only  with  tlie  small  hypnthetkal  di.splacement  winch  wo 
sjive  the  point  of  application  of  the  force  in  applying  the  principle  of 
virtual  velocitit«.  But  in  Kinetics  ttie  bodies  are  in  motion  ;  the 
f.)rce  (ictiKdli/  disjjlaces  its  iioint  of  ajiplication  in  such  a  manner  that 
the  displacement  has  a  projection  alonfr  the  direction  of  the  force.  If 
(1.1  denote  the  projection  of  any  elementary  arc  of  a  curve  alonjr  the 
direction  of  P,  the  work  done  by  /'  in  this  dis))lncement  is  /'(/,v.  The 
sum  of  all  these  elements  of  work  done  by  /'  in  its  mmion  over  a 
finite  space  is  the  whoh;  work  found  by  taking  the  integral  of  Pds 
between  proper  liinit.s. 

Hence  generally,  if  »  be  an  arc  of  the  path  of  a  particle,  P  the 
tangential  component  of  the  forces  whicli  act  on  it,  the  work  done  on 
the  particle  betv  ecu  any  two  points  of  its  path  is 

/Pih,  (1) 

the  inteirral  being  taken  between  limits  corresponding  to  the  initial 
and  (ina)  positions  of  the  jiarticle. 

213.  Work  on  an  Inclined  Plane.— Let   «  he   the 

inclination  of  the  plane  to  the  horizon,  W  the  weight 
moved.  .V  the  di.stance  along  the  plane  through  which  the 
weisrht  is  moved.  Ifesolve  W  into  two  comi)onents,  one 
along  the  plane  and  the  other  perpendicuhir  to  it ;  the 
former,  W  sin  «,  is  the  comjionent  which  resists  motion 
along  the  plane.  Hence  the  aiiKJunt  of  work  re<piired  to 
draw  the  weiglit  up  the  i)lano  =  IT  sin  a  •  s  =  U'xthe 
vertical  height  of  the  plane  ;  /.  c,  the  anion )if  of  work 
required  is  iinchnxtjed  by  Ihe  fubstituiion  of  the  oblique  pnfh 
for  the  vertical.  Ileiire  the  irork  in  monnf/  a  tuxlij  up  an 
inclined  plane,  without  friction,  is  equal  to  the  product  of 
the  veifjht  of  the  body  by  the  vertical  hciijht  through  which 
it  is  raised. 


'.'lUi  WOKK   O.V   .1  V    l.\i'LI.\t:n    VLAXh. 

Coit.  1. — If  tlie  \i\m\k'  1)0  roiijili,  let  /<  =  tlic  coeflicicnt 
of  friction  ;  thou  sini'o  tlic  iioriinil  coinijoiu'iit  of  the  weight 
is  II'  ciis  «,  tlie  resistiiiice  of  frietioii  is  /t  IT  cos  »<  (Art.  !)•»'). 
The  work  nM|uire(l  consists  of  two  parts,  (1)  raising  tiic 
weight  along  the  plane,  and  {-l)  overcoming  the  resistance 
of  IVietion  along  the  plane,  tlio  former  =  II' sin  «  •  s.  ami 
tile  latter  is  //  il  cos  ,£ .  x.  Hence  tlie  whole  work  ni'ces.s(iri/ 
to  more  lite  iveight  up  I  he  plane  is 


(sin  «  +  fi.  cos  «)  s. 


0) 


Since  s  sin  <c  represents  the  vertical  height  through 
which  the  weigiit  is  raised,  and  .v  cos  «  the  horizontal  si)ace 
througii  which  it  is  drawn,  this  result  may  he  stated  thus  : 
The  work  expended  is  the  satne  as  that  which  would  be 
required  to  raise  the  weight  throin/h  the  vertical  height  of 
the  plane,  together  with  that  which  would  be  required  to 
draw  the  t>ody  along  the  base  of  the  plane  horizontallif 
against  friction, 

Coii.  2. — If  a  body  be  dragged  through  a  spare,  s,  down 
an  inclined  plane,  whirh  is  ton  rough  for  the  body  to  slide 
down  by  itself,  the  work  done  is 


ir(/t  cos  «  —  sin  «)  s. 


(2) 


Cor.  3. — If  h  =  the  height  of  the  inclined  plane,  and 
b  —  its  horizontal  hase,  then  the  work  done  against  gravity 
to  move  the  body  u])  the  plane  =  Wh  ;  and  the  work  done 
against  friction  to  move  tlie  body  along  the  plane,  suppos- 
ing it  to  be  horizontal,  =  tibW.  Hence  (Cor.  1)  the  total 
work  done  is 

]\7i  +  libW.  (;{) 

If  the  body  be  ilrawn  down  the  ])lane,  the  total  work 
expended  (Cor.  'i)  is 

—  117/  -t-  ld>W.  (4) 


XE. 


hXAMJ'LKS. 


39:5 


=  tlic  coeflicicnt 
'lit  of  the  weight 
■  cos  «  (Art.  !)•.>). 
(1)  raising  tlic 
g  the  resistance 
^^sill  «  •  s,  and 
)  work  necessary 


(1) 

height  through 
horizontal  space 
lie  stated  thus  : 
whicli  wotihf  hi 
rlical  heufht  of 
I  be  rei/uirrd  to 
me  horizonlallt, 


a  space,  s.  down 
he  body  to  slide 


(3) 

incd  jilane,  and 
e  against  gravity 
I  tiie  work  done 
3  plane,  snppos- 
jor.  1)  the  total 

(3) 

the  total  work 


If  in  (4)  the  former  term  is  greater  than  the  latter, 
gravity  dues  more  work  than  what  is  expended  on  friction, 
and  tiie  body  slides  down  tiie  plane  with  accelerated 
velocity. 

.Sen.  1. — If  tiie  inclination  of  the  plane  is  small,  as  it  is 
in  most  cases  wiiieli  occur  in  practice,  as  in  coninion  roads 
and  railrorjls,  cos  «  may  without  any  important  error  be 
taken  as  e(jual  to  unity,  and  the  exjjression  for  tlie  work 
becomes  (C'ors.  1  and  'Z) 


W  {(IS  ±  s  sin  «), 


{^) 


the  upper  or  lower  sigu  being  taken  according  as  the  body 
is  dragged  up  or  down  the  plane. 

Sen.  2. — If  tlie  inclination  of  the  plane  is  small,  as  in 
the  case  of  railway  gradients,  the  j)ressure  ujion  the  idane 
will  lie  very  nearly  e(jnal  to  the  weight  of  the  body;  and 
the  total  work  in  moving  a  body  along  an  inclined  plane 
will  be  from  (3)  and  (4), 


[ilW±  Wh, 


(6) 


where  filW  is  the  work  due  to  friction  along  the  plane 
of  length  I,  and  Wh  is  the  work  due  to  gravity,  the  proper 
sigu  being  taken  as  in  (5). 

EXAMPLES. 

1.  How  much  work  is  done  in  lifting  150  and  200  lbs. 
through  the  heights  of  80  and  120  ft.  respectively. 

The  work  done  =  150  x  80  +  200  x  120 
=  lUJOOO  foot-pounds,  Ans. 

2.  A  body  weighing  500  lbs.  slides  on  a  rough  horizontal 
plane,  the  coeUicieiit  of  friction  being  0.1 ;  how  niuch  work 
must  be  done  against  friction  to  move  the  body  over 
100  ft.  ? 


l!Mto«i 


394 


KXAMI'LKS. 


Hero  tlu'  fruition  is  ii  force  of  '>()  lbs.  acting  directly 
opposite  to  tin-  molion  ;  liciice  the  work  done  iigainst  fric- 
tion to  move  the  hody  over  UM)  ft.  is 

50  X  100  =^  5000  fbot-pou?uL<,  Ans. 

'^.  A  train  weighs  100  tons;  the  total  resistance  is  8  lbs. 
per  ton;  how  much  work  must  be  expended  in  raising  it 
to  the  top  of  an  inclined  plane  a  mile  long,  the  inclination 
of  the  ])lane  being  1  verticid  to  70  horizontal. 

Hero  the  work  done  against  friction  (Sell.  2) 

=  800  X  5280  =:  4224000  foot-pounds, 
and  the  work  done  against  gravity 

=  224000*  X  5280  X  Vo  =  1089G000  foot-pounds, 
BO  that  the  whole  work  =  21120000  foot-pounds. 

4.  A  train  weighing  100  tons  moves  30  miles  an  hour 
along  a  horizontal  road;  the  resistances  are  8  lbs.  per  ton; 
find  the  quantity  of  work  expended    ach  hour. 

A)is.   12G720000  foot-pounds. 

5.  If  25  cubic  feet  of  water  are  pumped  every  5  minutes 
from  a  mine  t40  fathoms  deep,  recjuired  the  amount  of 
work  expended  per  minute,  a  cubic  foot  of  water  weighing 
02i  lbs.  Aiis.  202500  foot-pounds. 

C.  How  much  work  is  done  when  an  engine  weighing 
10  tons  moves  half  a  mile  on  a  horizontal  road,  if  the 
total  resistance  is  8  lbs.  per  ton. 

Ans.  211200  foot-pounds, 

7.  If  a  weight  of  1120  lbs.  be  lifted  up  by  20  men,  20  ft. 
high,  twice  in  a  minute,  how  much  work  does  each  man 
do  i)er  hour  ?  Ans.  134400  foot-pounds. 

»  Oii«  ton  boing  -JilO  lbs.  iiiiIokk  cilbcrwlHC  Ktatcd. 


HOUSE  POWBN. 


;5!t:. 


■ting  directly 
against   tVic- 

iind.;!,  Anfi. 

iince  is  8  lbs. 

in  raising  it 

10  inclination 


nds, 

)t-pound8, 

lids. 

rtiles  an  hour 
5  lbs.  per  ton  ; 
r. 
;(»ot-pounds. 

cry  5  minutes 
he  amount  of 
^ater  weighing 
'oot-pouuds. 

gine  weighing 
1   road,   if  tiie 

toot-pounds. 

20  men,  20  ft. 
does  each  man 
'oot-ponnds. 

ted. 


8.  A  l)ody  falls  down  the  whole  length  of  an  ineiin.'i 
plane  on  which  the  coetHcient  of  friction  is  0.2.  'i  hi' 
lieight  of  tlie  plane  is  10  ft.  and  tlie  l)ase  ;50  ft.  On  reach- 
ing the  bottom  it  rolls  horizontally  on  a  plane,  having  the 
same  coefficient  of  friction.     Find  how  far  it  will  roll. 

Ann.  20  ft. 

9.  IIow  nuicli  work  will  be  required  to  pumj)  8000  cul)i(' 
feet  of  water  from  a  mine  whose  depth  is  500  fathoms. 

Ans.   1500000000  fool-pounds. 

10.  A  hor.<e  draws  loO  lbs.  out  of  a  well,  by  means  of  a 
Yo\w  going  over  a  Hxed  jiulley,  moving  at  the  rate  of 
'2\  miles  an  hour;  how  many  units  of  work  does  this  horse 
perform  a  minute,  neglecting  friction. 

Ans.  ;53O0O  units  of  work. 

214.  Horse  Pcwer. — It  would  be  inconvenient  to 
ex])re,ss  the  power  of  an  engine  in  foot-pounds,  since  this 
unit  is  so  small ;  the  term  Horse  Power  is  therefore  u.^ed 
in  measuring  the  performance  of  steam  engines.  From 
experiments  made  by  Boulton  and  Watt  it  was  estimated 
that  a  horse  could  raise  ;}3000  lbs.  vertically  through  one 
foot  in  one  minute.  This  estimate  is  ])robably  too  high  on 
the  average,  but  it  is  still  retained.  Whether  it  is  greatei- 
or  less  than  the  ])ower  of  a  horse  it  matters  little,  while  it 
is  a  |)ower  so  well  defined.  A  Horse  Power  therefore  means 
(I  power  which  can  perform  33000  foot-pori nds  of  work  in  a 
minute.  Thus,  when  we  say  that  the  actual  horse  power 
of  an  engine  is  ten,  we  mean  that  the  engine  is  able  to  per- 
form 3;}0000  foot-pounds  of  work  per  minute. 

It  has  boeii  ostinintcil  that  %  of  the  33000  fool  iwunds  would  be 
about  the  work  of  a  horse  of  averatje  stronprtli.  A  mule  will  jierform 
I  the  work  of  a  horse.  .\ii  asH  will  ])erforin  al)out  \  the  work  of  a 
horse.  A  man  will  do  about  /,,  the  work  of  a  liorse,  or  about  ;f3()0 
units  of  work  per  minute.  Seo  Evers'  Ai)|)lied  Mech's;  also  Byrne's 
Practical  Mech's. 


B^^ 


;ii)(>        n^OItK   OF  L'AfSI.Ml    A    SVSTKM    o/     \Yi:i(IIITS, 

215.  Work  of  Raising  a  System  of  Weights.— 

Ijct  /',  Q,  li,  1)0  aiiv  tl>n.'i'  wciglils  at  iIr-  (li.staiici's,  y.  y, 
/•,  rc^poctivc'ly  alxivoa  tixinl  Imrizoiital  plaiu'.  'I'lu'ii  |.\rt. 
r)9  (3)]  or  (Art.  T3.  Cor.  3),  tlic  distaiKv  uf  tlie  foiitrc  i.f 
gravity  of  I',  Q,  R,  above  tlii.s  lixod  horizontal  i)laiic  is 


Pp+Qq  +  Rr 
I'-f  (J  +'R    ' 


(1) 


Now  suppose  that  the  weights  are  raised  vertieally 
through  the  heights  a,  b,  c,  resj>eetively.  Then  the  dis- 
tance of  the  centre  of  gravity  of  the  three  weights,  in  the 
now  jjosition,  above  the  .same  fixed  horizontal  i)lano  is 


f  +  Q+  R 

Subtracting  (1)  from  (••i),  wo  have 

7'  +  \f  +  R    ' 


i'i) 


(^) 


for  the  vertical  distance  between  the  two  po-sitions  of  the 
centre  of  gravity  of  the  three  bodies. 

Now  the  work  of  raisii'g  vertically  a  weight  e(|ual  to  the 
sum  of  /-*.  Q,  R.  through  the  s]»ace  denoted  by  (3)  is  the 
product  of  the  sum  of  the  weights  into  the  space,  which  is 


Pa  +  Qb.+  Rr, 


(-t) 


but  (4)  is  the  work  of  raising  the  throe  weights  /\  Q,  R, 
through  the  heights  n,  h,  c,  respectively.  In  the  sanio  way 
this  may  be  shown  for  any  number  of  weights. 

Hence  when  several  weights  are  raised  vertically  through 
different  heights,  the  whole  work  done  is  the  same  as  that  of 
raising  a  weight  equal  to  the  sum  of  the  weights  vertically 
from  the  first  position  of  their  centre  of  gravity  to  the  last 
position.     (See  Todhunter's  ^kch"s.  p.  33S.) 


•KUITS. 

Weights.— 

listaiiccs.  //.  7. 
.  Tlion  [Art. 
'  the  ft'iitn"  of 
ill  plane  is 

(1) 

ised  vertically 
Then  the  clis- 
ivoiglits,  in  the 
1  ])lane  is 


-^•). 


(2) 


(3) 

lositions  of  the 

ht  e(|ual  to  tiie 
il  by  (:{)  is  the 
space,  which  is 

(4) 

Mghts  /',  Q,  R, 
11  the  saiiu'  way 
its. 

'•rticalUf  through 

samp  as  thai  of 

'<'i(/hts  vertically 

ivitii  to  tho  Iftst 


EXAMPLES. 


EXAMPLES. 


3;i7 


1.  How  manv  horse-pnwer  would  il  lake  to  raise  '^  cwt. 
of  coal  a  minute  from  a  i)it  whoso  depth  is  110  fathoms';' 

Depth  =  110  X  'i  =  <ifiO  feet. 

3  cwt.  =  11-^  X  :i  =  :33(;  Ihs. 

Hence  the  work  to  he  done  in  a  minute 

=  GGO  X  33G  =  --i-^lTfiO  loot-pounds. 

Therefore  the  horse-i)o\ver 

=  -Z-ZVitO  -=r  33000  =  i).ri.  Am. 

3.  Find  how  many  cubic  iVcl  of  waier  an  engine  of 
40  horse-jtower  will  raise  in  an  hour  from  a  mine  80 
fathoms  deep,  supposing  a  cubic  loot  of  water  to  weigh 
1000  ozs. 

Work  of  the  engine  per  hour  =z  40  x  33000  x  60  foot- 
pounds. 

Work  expended  in  raising  one  cubic  foot  of  water 
through  80  fathoms  —  'H"  >'  *^"  x  '^  =  '^*^^'^'*  ^"o^" 
pounds. 

Hence  the  number  of  cubic  feet  raised  in  an  Inur 

=  40  X  33000  X  00  ^  30000  =  :iG40,  Ans. 

3.  Find  the  horse-power  of  an  engine  which  is  to  move 
at  the  rate  of  20  miles  an  hour  up  an  incline  which  ri.ses 
1  foot  in  100.  the  weight  of  the  engine  and  load  b"ing 
GO  tons,  and  the  resistance  from  friction  I'-i  lbs.  per  ton. 

The  horizontal  space  ])asscd  over  in  a  minute  =  ITGOft.; 
the  vertical  space  is  one-lumdredth  of  this  rr:  17. GO  ft. 
lieiice  from  (G)  of  Art.  213,  we  have 

12x1  rOO  X  GO  +  GO  X  2240  x  1  T.G  =  1 TGO  x  20G4  foot-pounds. 


398  EXAMl'LES. 

Therefore  the  horse-power 

=  ir<iO  X  -MM  -r-  :3:}(lO()  =  110.(18,  J  Hi. 

4.  A  well  is  to  be  dug  'iU  I't.  deei),  and  4  ft.  in  diameter: 
find  the  work  in  raising  the  niuterial,  suppo.sing  tiiat  a 
cnhie  foot  of  it  weighs  140  Ib.s. 

Here  the  weight  of  tl)e  material  to  be  raised 

=  4Tr  X  ;J0  X  140  =  140  x  80t  lbs. 

The  work  done  is  equivalent  to  raising  this  through  the 
height  of  10  ft.  (Art.  ■■llr>).     Hence  the  whole  work 

=  140  X  80tt  X  10  —  ]12000tt  foot-pounds,  Jh,s. 

5.  Find  the  horse-power  of  an  engine  that  would  raise 
7'  tons  of  coal  per  hour  from  a  pit  whose  depth  is  a 
fathoms. 

Work  jKT  minute  = =  22iaT; 

•.     the  horse-power  =:   ..7—-- ,  Ans. 

O'iOOU 

(!.  Re<[uired  the  work  in  raising  water  from  three  different 
levels  whose  deptlis  are  a,  h,  c  fathoms  respeetively  ;  from 
the  first  J,  from  the  second  B.  from  the  third  C,  cubic 
feet  of  water  are  to  be  raised  per  minute. 

Work  in  raising  water  from  the  first  level 

=  G2.5  J  X  «  X  G  =  375  J. «; 

and  so  on  for  the  work  in  the  other  levels  ; 

.■.     work  per  min.  =  375  {A-(i+  li-h-\-  C-c)  foot-pounds. 

7.  Fin  i  tlu^  horse-power  of  an  engine  which  draws  a 
load  of  T  tons  along  a  level  rouii  at  the  rate  of  m  miles 


Ans. 

.  ill  diameter : 
posiuf^  tliat  a 

i 

lbs. 

s  through  the 
3  work 

iidfi,  Ans. 

lit  would  raise 
)se  depth  is  a 

=  2UnT; 


three  different 
.'ctivcly  ;  from 
hird  C,  cubic 


foot-pounds. 

hich  draws  a 
le  of  in  mik's 


EXAMPLES. 


399 


an  hour,   the  friction    Ijeing  />  pounds  per  ton,  all  other 
resistances  being  neglected. 
Work  of  the  engine  per  minute 

_     5380  ?»  „T    ,n 

=  Tp  --^^p-  =  88  Tpm. 


~    33000    ~    3000   ' 


ns. 


8.  Required  the  number  of  horse-power  to  raise  2200 
cubic  ft.  of  water  uii  hour,  from  a  mine  whose  depth  is  03 
fathoms.  Ans.  2G^. 

9.  What  weight  of  coal  will  an  engine  of  i  iiorse-power 
raise  iu  one  hour  from  a  pit  whose  depth  is  200  ft.  ? 

Ans.  39G00lbs. 

10.  In  what  time  will  an  engine  of  10  horse-power  raise 
5  tons  of  material  from  the  depth  of  132  ft.? 

Ans,  4-48  minutes. 

11.  How  many  cubic  feet  of  water  will  an  engine  of  36 
liorse-power  raise  in  an  hour  from  a  mine  whose  depth  is  40 
fathoms  ?  Ans.  4752  cubic  feet. 

12.  The  piston  of  a  steam  engine  is  15  ins.  in  diameter  ; 
its  stroke  is  2^  ft.  long ;  it  makes  40  strokes  per  minute ; 
tlie  mean  pressure  of  the  steam  on  it  is  15  lbs,  per  square 
inch;  what  number  of  foot-pounds  is  done  by  the  steam 
per  minute,  and  what  is  the  iiorse-jjower  of  the  engine  ? 

Jmv,  205072.5  foot-pounds  ;  8-03  11.-?. 

13.  A  wciglit  of  1^  tons  is  to  be  raised  from  a  depth  of 
50  fath(jms  in  o'le  minute;  determine  the  horse-power  of 
the  engine  capable  of  doing  the  work. 

Ans.  30t8j  IT,-J». 


♦  T'.ie  IetttT«  n.-P.  arc  often  used  a»  ftbbrcvlatlone  of  the  words  horse-power. 


ita 


400 


MODILUS   OF  A    MACHINE. 


14.  The  resisUince  to  the  motion  of  a  certain  body  is 
440  lbs.;  how  many  foot-ijouiids  imisl  be  expended  in 
making-  this  body  move  over  30  miles  in  one  hourV  What 
nuist  be  the  horse-power  of  an  en^riiie  that  does  the  same 
naniber  of  foot-pounds  in  tiie  same  tinie!-' 

Ans.   ti'JG'JGOOO  foot-pounds;  ;)5|  II.-P. 

If).  An  eii'dne  draws  a  load  of  '!()  tons  at  the  rate  of  20 
miles  an  hour:  the  resistances  are  at  tlie  rate  of  8  lbs.  ))cr 
ton  ;  iiiid  the  horse-i)Ower  of  the  engine.  Aux.   ^o-O. 

IC).  How  many  cubic  feet  of  water  will  an  engine  of  'IhO 
horse-power  raise  per  minute  from  a  depth  of  2oo  fathoms":' 

A)iK.    110  cubic  ft. 

17.  Til  ere  is  a  mine  with  three  shafts  which  are  respec- 
tively 300,  450,  and  oOO  ft.  deep:  it  is  re(|uired  to  raise 
from  the  first  SO.  from  the  second  00.  from  the  third  40 
cubic  ft.  of  water  per  minute;  liud  the  horse-i)ower  of  thi' 
engine.  -'■'•>•■•   1'54||. 

216.  Modulus*  of  a  Machine.— The  wliole  work  per- 
formed by  a  machine  consists  of  two  parts,  liie  iificfid  work 
and  the  lod  work.  Tiie  useful  work  is  that  whicii  the 
maciiine  is  designed  to  produce,  or  it  is  liiat  whieli  is 
employe'^  in  overcoming  u.^cfid  resistances  :  the  lost  work 
is  thai  which  is  not  wanted,  but  wiiich  is  uiuivoidai)ly 
produced  or  it  is  that  which  is  spi'nt  in  overcoming  wnxtc- 
fill  resistances.  For  instance  in  drawing  a  train  of  cars,  tho 
useful  work  is  ])erformed  in  niov.ng  the  train,  but  the  lost 
work  is  tliat  which  is  doito  in  overcoming  tho  friction  of 
the  train,  the  resistance  of  gr;i\  ity  on  up  grailes.  the  resist- 
ance of  the  air,  v\r.  'I'lie  woi'k  iipplied  to  a  machine  is 
('(puii  to  (he  wiiol.'  work  done  l»\  tiie  nuichine.  botii  useful 
and  lost,  liierefc'v  the  useful  work  is  always  less  than  the 
work  applieil  to  tiic  machine. 


*  Sonii'tliufi*  tttlli'd  Kmcli'iicy.    (Art.  108.) 


KXAMI'LES. 


401 


tain  body  is 
cxpeiuled  in 
lOurV  What 
)L'S  the  same 

35|  II.-P. 

e  rate  of  ^0 
of  8  ll)s.  ))cr 
Iw.v.   ro-i). 

n<!^ine  of  ^.'lO 
2(10  tiitlioms":' 
0  cubic  ft. 

h  lire  rcspce- 
irod  to  riiise 
tlie  tiiird  40 
power  of  tbe 
■'ts.  i:54||. 

lie  work   per- 

Hscfiil  work 
lit  wiiieii  tlie 
iiat  \vl)ieli  is 
he  lo.st    work 

unavoidal)l_v 
iominj;  trns/c- 
in  of  cars,  the 

l)ut  the  lost 
ho  friction  of 
L's,  tlie  resist- 
M  machine  is 
'.  botii  nsel'nl 

less  than  the 


The  Modulus  of  a  mdi-hine  is  the  ratio  of  tho  useful  work 
(lour  to  tlie  work  ap/ilieil.  Thus,  if  the  work  applied  to  an 
engine  be  40  horse-power,  and  the  engine  delivers  only  .'30 
horse-i)ower.  the  modulus  is  J,  /.  e.,  one-quarter  of  the  work 
apj)lied  to  the  machine  is  lost  by  friction,  etc. 

Let  ir  i)e  the  work  applied  to  the  machine,  ll'„  the  use- 
ful work,  and  ui  the  modulus.  Then  we  have  from  the 
above  definition 

W 

"'  =  T,;-  (1) 

If  a  machine  were  perferl.  i.  /'.,  if  there  wore  no  lost  work, 
the  modulus  would  l)e  unity;  but  in  every  machine,  some 
of  the  work  is  lost  in  overcoming  wasteful  resistances, 
so  that  tile  modulus  is  always  less  than  unity  ;  and  it  is  of 
course  the  object  of  inventors  and  improvers  to  bring  this 
fraction  as  near  to  unity  as  possible. 

EXAMPLES. 

1.  An  engine,  of  X  effective  ''.jrsc-power,  is  found  to 
pump  A  CI  'lie  ft.  of  water  per  isiin..  from  a  mine  a  fathoms 
ileep  ;  lind  the  modulus  of  the  ])iimps. 

Work  of  the  engine  per  min.  —  ;{:{0()0  .V  II. -P. 

'i'he  useful  v,'ork,  or  work  exju'iidcd  in  pumping  water, 

=  (VZ-'y  A  X  <)''  =  -JT")  A- a; 

henc(  Trom  (1 )  wc  have 

'^  -  ;):iOooA^~  88  ,V'  '"'''• 

■.*.  There  were  .\  cul)ic  ft.  of  water  in  a  mine  wlic-e  deptli 
is  u  fathoms,  when  an  I'Ugiiu'  of  .V  horse-power  began  to 
work  liie  pumi»:  the  water  c<»ntinued  (o  ilow  into  the  mine 
at  the  rate  of  U  cubic  ft.  per  minutej  required  (he  (inu; 


mmm^m 


t^ 


402 


EXAMPLES. 


in  which  the  mine  would  l)i'  cleiirod  of  water,  the  niodnhis 
of  the  pump  tieiug  iii. 

Let  X  =  the  number  of  minutes  to  clear  the  mine  of 
water.     Then 

weight  of  water  to  be  pumped  =  (J:.; -5  (A  +  Bar)  ; 
work  in  pumping  water  =  37r)rt  (A  +  B.r)  foot-pounds; 
effective  work  of  the  engine  =  m-  Ji-'.i'MOOx  ; 
.  • .    a.'iOOO  m  Nx  =  3?5rt  (A  +  lix)  ; 
A- a 


X  =z 


88m.V—  h-a' 


Ans. 


3.  An  engine  has  a  0  foot  cylinder ;  the  shaft  makes  30 
revolutions  per  minute,  the  average  steam  pressure  is  25 
lbs.  per  H(iuarc  inch  ;  recjuired  tiie  hoise-powcr  wiien  the 
area  of  the  jjiston  is  1800  square  inches,  t lie  modulus  of 
the  engine  being  [J. 

Work  done  in  one  minute  =  IHOO  x  ^"»  x  'J  X  2  x  30 
foot-ponnds.  We  multiply  by  twice  the  length  of  the 
stroke,  because  the  piston  is  driven  both  u|>  and  down  in 
one  revolution  of  the  shaft. 

Tlie  effective  Jiorse-power  =  isiLa^n^iu  lu  x  [J 

=  iM,   A  U.S. 

4.  The  diameter  of  the  j)isloii  of  a  steam  engine  is  (10 
ins.;  it  makes  11  strokes  per  minute;  the  length  of  each 
stroke  is  8  ft.;  the  mean  pressure  per  s(piare  in.  is  1.5  ll)s.; 
reipiired  the  luimlier  of  cubic  ft.  of  water  it  will  raise  per 
hour  from  a  depth  of  50  fathoiiLs,  tlie  modulus  of  the 
engine  l)eing  O-tl.'"). 

The  nuinbcr  of  l"()i)t-|Hmii<lH  of  useful  work  done  in  one  lioui' iiml 
■[i^nt  in  raining  water  =^  ir  ^  80'  >  8  x  in  •  tl  x  (to  .  o  ().">.  tlieiefoi'<',  etc 

Ans.  7703  cubic  ft. 


f 


he  modulus 
lie  mine  of 

+  Bx) ; 

)t-pouuds ; 
)00x; 


rt  makes  30 
I'ssurc  IS  25 
r  wlioii  the 
•moduhis  of 

<;  X  2  X  130 
i^ftli  of  the 
iiul  down  iu 


'tifriuc  is  (10 
ii;tli  ot'eueh 
I.  is  15  lbs.: 
iill  raise  per 
luhis  of  tiie 


iiiii'  liiiiii'  mill 

tlitl'cfiu'c,  etc 

I  cubic  ft. 


f 


KXAMPLEH. 


iO-.i 


i).  An  engine  is  required  to  j)umi)  1000000  gallons  of 
water  every  \'i  hours,  from  a  mine  13'^  fathoms  deep  ;  find 
the  horse-power  if  the  modulus  be  \^,  and  a  gallon  of 
water  weighs  10  lbs.  Ans.  SGSy'ij  II.-P. 

(i.  What  must  be  the  horse-power  of  an  engine  working 
e  hours  per  day.  to  suj)j)ly  ii  families  with  //  gallons  of 
water  each  pm-  day.  siij)posing  the  water  to  be  raised  to  the 
mean  height  of  //  feet,  uiu!  that  a  gallon  of  water  weighs  10 
lbs.,  the  modulus  being  in.  lujh        ,,    ., 

""'•  nt8ooo7w        ■ 

T.  Water  is  to  be  raised  from  u  mine  at  two  dilTerent 
levels,  viz.,  50  and  80  fathoms,  from  the  former  30  cubic  ft., 
and  from  the  latter  15  cubic  ft.  per  minute  :  find  the  horse- 
l)o\ver  of  the  machinery  that  will  be  required,  assuming 
the  modulus  to  be  O-O.  Ans.  51- U  II.-P. 

8,  'Che  diameter  of  the  piston  of  an  engine  is  80  ins.,  the 
mean  pressure  of  the  steam  is  Vl  lbs.  |)er  square  inch,  the 
length  of  the  stroke  is  10  ft.,  the  number  of  strokes  made 
per  minute  is  1 1  ;  how  many  cubic  ft.  of  water  will  it  raise 
))er  minute  from  a  depth  of  250  fathoms,  its  modulus  being 
0.0?  Ans.  42-40  cubic  ft. 

II.  If  the  engine  in  the  last  example  had  raised  65  cubic 
ft.  of  water  per  minute  from  a  depth  of  250  fatlumis,  what 
would  have  been  its  modulus?  Ans.  0-7TT1. 

10.  llow  many  strokes  jter  minute  must  the  engine  \\\ 
Ex.  8  make  in  order  to  raise  15  cubic  I't.  of  water  per 
minute  from  the  given  depth?  J«.v.   4. 

11.  What  must  be  the  length  of  the  stroke  of  an  engine 
whose  modidiis  is  (I- 05,  and  whose  other  dimensions  and 
eouilitinns  of  working  are  I  he  same  as  in  l']\.  S,  if  thev  both 


do  the  same  quanlity  of  useful  work? 


I  us. 


!3  ft. 


^ 


404 


Kl.VETIC   AM)    fOTEyriAL    KSKlidV. 


217.  Kinetic  and  Potential  Energy.  Stored 
Work. —  The  viwrgy  of  a  body  mmiis  i/.<  jxi/rrr  of  doitui 
work;  and  tlic  total  amount  of  encrqy  possessed  t>i/  I  lie  body 
■is  mmsKicd  by  tlie  total  amount  of  work  wldrl,  it  /,s-  rapable 
of  doiny  in  j'nssiny  from  its  prm'nl  condition  to  some 
standard  condition. 

Every  niuving  Ixuly  po.ssessi'.s  energy,  tor  il  can  he  made 
to  do  work  l)y  parting  with  its  veloeity.  'I'lie  velocity  of 
rlie  l)ody  may  l>e  used  for  causing  it  to  ascend  vertically 
against  the  attraction  of  the  earth.  /.  >:.  to  do  work  against, 
the  resistance  of  gravity.  A  caniioi!  hall  in  motion  can 
penetrate  a  resisting  hody;  water  flowing  against  a  water- 
wheel  will  turn  the  wheel  ;  the  moving  air  drives  the  ship 
through  the  water.  Wherever  we  find  matter  in  motion 
wo  have  a  certain  amount  of  eiiergv. 

Energy,  as  known  to  us,  hdongs  to  one  or  the  other  of 
two  cla.sses,  to  which  tiu'  mimes  kinetic*  cneryy  and 
potential  I'neryy  are  given. 

Kinetic  enen/y  is  eneryy  tliat  a  body  possesses  in  virtue  of 
its  beiny  in  motion.  It  is  energy  actually  in  use,  energy 
that  is  constantly  i)eing  spent.  'I'lie  energy  of  a  hiiUet  in 
motion,  or  of  a  ily  wheel  revolving  raj)idly,  or  of  a  pile- 
driver  just  hefore  it  strikes  the  pile,  are  examples  of  kinetic 
cneryy.  The  work  done  hy  a  force  on  a  hody  free  to  move, 
exerted  through  a  given  distance,  is  always  ('(jI'mI  to  the 
C(?rrespoiiding  increase  of  kinetic  energy  [Art.  ISD  (;i)].  If 
a  mass.  /«,  is  moving  with  a  velocity,  r.  its  kinetic  energy 
is  pnr'^  [(;!)  of  Art.  18il].  If  this  velocity  ho  generated  hy  a 
constant  force,  P,  acting  through  a  space,  ,s,  we  have. 
(Art.  211) 

Ps  ^  hin^,  (1) 

thai  is,  the  work  done  on   the  hody  is  the  exact   e(|iiiTali'nl 
of   the    kinetic   energy,    and    the    kim'tie  energy  is  rwon- 

♦  Ciillwl  iiIho  (uiuut  (nenjij,  di  (iieryij  (tf  motion. 


II  y. 


KISKTIV  AM)   POTKXriAL   E.\EROV. 


45 


jy.       Stored 

miriT  of  (loiiKj 
'd  hji  the  boihj 
'i  it  is  f((pallc 
it  ion    to   tiome 


can  he  made 
'lie  velocity  of 
eml  vertically 

work  iiyaiiist 
II  motion  can 
linst.  a  water- 
'ives  the  ship 
;er  in  motion 

■  the  other  of 
cmrtjji    and 

s  in  virtue  of 
in  use,  energy 
)f  a  bnllet  in 

or  of  a  pile- 
)les  oi"  k-in('tic 
free  to  move, 

e(|i"iil  to  the 
.  1^!»  (:i)l.  If 
inelic  energy 
;eiieralod  by  a 

.s,    we   have. 

(I) 

ct  c(|iiiTali'nl 
Tgy  is  riH'on- 


vertiblo  into  the  work;  and  the  exact  amonnt  of  work 
which  the  mass  m,  witii  a  velocity  v,  can  do  against  resist- 
ancu  before  its  motion  is  completely  destroyed  is  hin\ 
'I'liis  is  called  xtoml  ivork,*  and  is  the  amount  of  work  that 
any  opposing  force,  /',  will  have  to  do  on  the  lunly  before 
bringing  it  to  rest.  'I'lius,  when  a  heavy  Jly-wheel  is  in 
rapitl  motion,  a  considerable  jwrtion  of  the  work  of  the 
-ngine  must  have  gone  to  produce  this  motion  ;  and  before 
the  engine  can  come  to  a  state  of  rest  all  the  work  stored 
up  in  the  lly-wheel,  as  well  as  in  the  other  parts  of  the 
machine,  must  be  destroyed.  In  this  way  a  lly-wheel  acts 
as  a  reservoir  of  work. 

If  a  body  of  mass  //;,  moving  through  a  space  .v,  change 
its  velocity  from  v  to  Cj  the  work  done  on  the  body  as  it 
moves  through  that  space,  (Art.  18it),  is 


\m{v^-r,% 


C'i) 


If  the  body  is  not  perfectly  free,  /.  v.,  if  there  is  one  force 
urging  the  body  on,  and  another  force  resisting  the  body, 
the  kinetic  energy,  hnv'^,  gives  the  excess  of  the  work  done 
by  the  former  force  over  that  done  by  the  latter  force. 
Thus,  when  the  resistance  of  friction  is  overcome,  the 
moving  forces  do  work  in  overcoming  this  resistance,  and 
all  the  work  done,  in  excess  of  that,  is  stored  in  the  moving 
mass. 

Potential  energy  is  eneriiy  that  a  body  possesses  in  virtue 
of  its  position,  i'he  energy  of  a  bent  watch-spring,  which 
does  work  in  uncoiling:  the  energy  of  a  weight  raised 
above  the  earth,  as  the  weight  of  a  clock  which  does  work 
in  falling  ;  the  energy  of  compressed  air,  as  in  the  air-gun, 
or  in  an  air-brake  on  a  locomotive,  which  does  work  in 
expamling;  the  energy  of  water  st.tred  in  a  mill-dam,  ami 
of  steam   in  a  boiler,  are  all  examples  of  potential  enertjy. 

*  I'allcd  «Ni)  mviimiitufeit  imrk.  S.u  Todluinli'i'H  Mi'chf.,  ulfo  wtorud  I'licrRy  anil 
DC  wo'k,     Diduho'm  Muclmiilt-',  \i.  178. 


ite 


40(5 


J-:XAMI'I.hS. 


Such  energy  may  ur  may  not  l)e  called  into  action,  it  may 
be  dormant  fur  years;  the  power  exists,  l)ut  the  action  will 
begin  only  when  the  weight,  or  tiie  water,  or  the  steam  is 
released.  Hence  the  word  potential,  is  significant,  at, 
expressing  that  the  energy  is  in  existence,  and  that  a  new 
power  has  been  conferred  upon  it  by  the  act  of  raising  or 
aonliiiing  it. 

For  example  suppose  a  weight  of  1  II).  be  projected 
vertically  ujjwards  with  a  velocity  of  '.i2-2  ft.  per  second. 
The  energy  imparted  to  the  body  will  carry  it  to  a  height 
of  16-1  ft.,  when  it  will  cease  to  have  any  velocity.  The 
whole  of  its  kinetic  energy  will  have  been  expended ;  but 
the  body  will  have  acquired  jtotential  energy  instead  ;  /.  c, 
the  kinetic  enei'gy  of  the  body  will  all  have  been  converted 
into  potential  energy,  which,  if  the  weight  bo  lodged  for 
any  time,  is  stored  up  and  ready  to  be  freed  whenever  the 
body  shall  be  permitted  to  fall,  and  bring  it  l)ack  to  its 
starting  point  with  the  velocity  of  IVi-  2  ft.  ])er  second;  ap.d 
thus  the  body  will  reacfpiire  the  kinet  c  energy  which  it 
originally  received.  Hence  kinetic  energy  and  potential 
energy  are  mutually  convert il)le. 

Let  h  be  the  height  through  which  a  body  must  fall  to 
ac(|uire  the  velocity  r,  in  and  IT  the  mass  and  weight, 
respectively.  Then  since  v^  =  'Zgh,  we  have,  for  the  stored 
work, 

W  W 

i""'  =  «,/  "'  =  •../  •  '^9l>'  =   "V^  (3) 


'i'.l 


•i'J 


Hence  we  may  say  that  the  work  stored  in  a  moving  body 
is  measured  hji  llic  prinlxrl  of  Ihc  iiwiiiht  af  the  hotli/  info  the 
heiijld  t/iroiiyh  ir/iic/i  il  niunl  fall  lo  (tcquire  the  velocity. 


EXAMPLES. 


1.   Let  a  bullet  leave  the  barrel  of  a  gun  with  the  velocity 
if  1000  ft.  per  second,  ami  suppose  it   to  weigh  2  ozs. ;  lind 


action,  it  may 
the  action  will 
)r  the  steam  is 
sigiiilioant,  us 
id  that  a  now 
ct  of  raising  or 

.   he   projected 

ft.  per  second. 

it  to  a  height 

velocity.    The 

expended ;  but 

instead  ;  i.  c, 

been  converted 

he  lodged  for 

whenever  the 

it  hack  to  its 

'r  second ;  ard 

lergy  which  it 

and  potential 

;ly  must  fall  to 
s  and  weight, 
for  tlio  stored 


(3) 

moving  hody 
",  Imtli/  into  the 
le  velocity. 


til  Ihc  velocity 
;li  'Z  ozs. ;  lind 


EXAMPLES. 


407 


the  work  stored  up  in  the  l)ullet,  and  the  height  from  which 
it  must  fall  to  acijvnre  that  velocity. 

liere  we  have  from  (3)  for  the  stored  work 

(1000)2  =  Wh 


2  X  IGg 

=  1941  foot-pounds. 
.'.    h  =  155;i8  feet. 

2.  A  ball  weighing  lo  lbs.  is  projected  along  a  horizontal 
piano  with  the  velocity  of  v  ft.  per  second  ;  what  space,  s, 
will  the  hall  move  over  before  it  comes  to  a  state  of  rest, 
the  coefficient  of  friction  being  /'? 

Here  the  resistance  of  friction  is  />,  which  acts  directly 
opposite  ti»  the  motion,  tberef(tre  the  work  done  by  friction 
while  the  body  moves  over  s  feet  =  fn's  ;  the  work  stored 

up  in  the  ball  =  f/wy"  =  —  ;  therefore  from  (1)  we  have 


tos  =  ., 


tvv* 


■'^y 


3.  A  railway  train,  weighing  7' tons,  has  a  velocity  of  v 
ft.  per  second  when  the  steam  is  turned  off ;  what  distance, 
,s,  will  tiie  train  have  moved  on  a  level  rail,  whose  friction 
is  p  ll)s.  i)er  ton,  when  the  velocity  is  i\  ft.  per  second  ? 

Here  the  work  done  by  friction  =  pTs;  lienco  from  {5J) 
wc  have 

_  U20  {o^  -  Vq^) 

•     •        « '     —  * 

f/P 

4,  A  train  of  T  dms  desccuids  iin  incline  of  s  ft.  in 
length,  havinvf  a  toliil  rise  <)f /H't.:  what  will  be  the  velocity, 
/',  acquired  by  the  train,  the  friction  Ijoing^y  lbs.  per  ton? 


ito 


408 


KIXKTIC   K.Xhh'l.y   Ot    A    h'KlIO   liODV. 


Here  we  have  (Art.  :213,  Scli.  •-.*).  Hie  work  done  on  tlio 
train  =  the  work  of  gravity  —  the  work  of  friction 

=  -iUi)  Tit  -pT.s; 

wliich  is  e(iiuil  to  the  work  stored  up  in  the  train.     Heuco 

22W  Tv^ 


^ 


=  2240  Th  —pTs; 


.'.    V  =  \/-l(jh  -  j^.ns[/ps. 

5.  If  the  velocity  of  the  train  in  the  last  example  ho 
y'o  ft.  per  second  when  the  steam  is  turned  otT,  what  will  he 
its  velocity,  v,  when  it  reaches  the  bottom  ol  the  incline? 

*;.  A  body  weighing  40  lbs.  is  projected  along  u  rough 
horizontal  plane  with  a  velocity  of  150ft.  per  sec;  the 
coctlicient  of  friction  is  \;  find  the  work  done  against 
friction  in  live  seconds.  Ans.  3500  foot-pounds. 

7.  Find  the  work  accumulated  in  a  body  which  weighs 
300  lbs.  and  has  a  velocity  of  (34  ft.  per  second. 

Ans.   19200  foot-pounds. 

218.  Kinetic  Energy  of  a  Rigid  Body  revolving 
round  an  Axis.— Let  m  be  tlie  mass  of  any  particle  of 
tiie  body  at  the  distance  r  from  the  axis,  and  let  w  be  the 
angular  velocity,  which  will  be  the  same  for  every  particle, 
since  the  body  is  rigid;  then  the  kinetic  energy  of  m  = 
\ni  (/•(.))«.  The  kinetic  energy  of  the  whole  body  will  be 
found  by  taking  the  sum  of  these  expressions  for  every 
particle  of  the  body.     Hence  it  may  be  written 


(1) 


lioitr. 

irk  (loiio  oil  tlio 
F  friction 


)  train.     Heuco 


last  cxamj)le  bo 
off,  what  will  be 
)i  tlio  inc'Iini'? 

//<  —  TTVff//;^'- 

I  along  a  rough 
k  per  sec. ;  the 
k  done  against 
)  foot-pounds. 

Jy  which  weighs 

:)nd. 

J  foot-pounds. 

ody  revolving 

any  imrtide  of 
id  let  6)  be  the 
r  every  particle, 
energy  of  m  = 
lie  body  will  be 
-sions  for  every 
ten 

(1) 


EXAMPLES. 


409 


i  tnr'^  is  called  the  moment  of  inertia  of  the  body  about  the 
axis,  and  will  be  explained  in  the  next  chapter. 

Hence  the  kinetic  energy  of  any  rottitiny  boily  =  ^I*^, 
where  I  is  Hie  moment  of  inertia  round  tlie  axiit,  and  w  tlie 
angular  velocity. 

In  the  ease  of  a  fly-wheel,  it  is  sutKcient  in  practice  to 
treat  the  whole  weiglit  a^^  distributed  uniformly  along  the 
circumference  of  the  circle  di'scribed  l)y  the  mean  radius 
of  the  rim.  Let  r  l)e  this  radius;  then  the  moment  of 
inertia  of  any  particle  of  the  wheel  =  mr'^,  and  the  moment 
of  inertia  of  the  whole  wheel  =  3/K  where  ^f  is  the  total 

mass.  Hence,  snbstiliiting  in  (1)  we  have  -— Mr\  which 
is  the  kinetic  energy  of  the  fly-wheel. 

EXAMPLES. 

1,  Two  equal  particles  are  made  to  revolve  on  a  vertical 
axis  lit  the  distances  of  a  and  b  feet  from  it;  required  the 
point  where  the  two  particles  uiiist  be  collected  so  that  the 
work  may  not  be  allt'ivd. 

Let  m  =  the  mass  of  each  particle,  k  =  the  distance  of 
the  refpiircd  point  from  the  axis,  and  w  =  the  angnlar  veloc- 
ity ;  then  we  have 

Work  stored  in  both  particles  =  J/h  (rtoj)2  +  ^m  (buyY; 

Work  stored  in  both  partick's  collected  at  j)oint  =  m  (ku)'', 

.■.■    m  (^•w)2  =  |/rt  {a(^Y  +  ^m  (iw)*; 

.-.     k  =  Vf(a^^+b^). 

This  point  is  called  the  centre  of  gyration.  (See  next 
chapter.) 

'i.  The  weight  of  a  fly-wheel  is  v>  lbs.,  the  wheel  makes 
H  revolutions  per  minute,  the  diameter  is  'ir  feel,  diameter 


m^ttmrn 


410 


KXAMl'l.t:S. 


of  axle  a  iiiclies,  ami  the  foetlicioiit  of  friction  on  tlie  axlo 
/;  how  many  revc'utions,  x,  will  the  wheel  make  before  it 
stops  ? 


AVork  stored  in  the  wlieel  =  "-  (  —  V/-2. 

•i(j  \  (JO  /    ' 

~  2y  Im" 
Work  done  by  friction  in  x  revolutions 


and  when  the  wlieel  stops,  we  have 


X  = 


3.  Required  tlie  number  of  strokes,  .'•,  which  tiie  fly-wheel 
in  the  last  example,  will  give  to  a  forge  hammer  whose 
weight  is  W  I'us.  and  lift  //  feet,  supposing  the  hammer  to 
make  one  lift  for  everv  revolution  of  tht;  wheel. 


Here  the  work  due  to  raising  hammer  =  Whx. 


.  &c 


W  7t2«2/'2 


4  It     It'll- 1  - 

Aii<3      qr   — - 


4.  The  weight  of  a  fly-wheel  is  SOOO  lbs.,  the  diameter 
yO  feet,  diameter  of  axle  14  inches,  eoellicicnt  of  friction 
0.2;  if  the  wheel  is  separated  from  I  lie  en;,nne  when  mak- 
ing 27  revolutions  per  ininute,  lind  bow  many  revolutions 
it  will  make  before  it  stoj)s  (y  taken  =  32.2). 

J«*'.  10.9  revolutions. 


on  on  the  axle 
make  before  it 


h  tlic  fly-wheel 
lammer  whose 
lie  hammer  to 
>cl. 

.  • .  &c 

li  '+  rrn/w) ' 

.  tlic  diameter 
'lit  of  friction 
111'  when  muk- 
iiy  revolutions 

revolutions. 


EXAMI'IjES. 


411 


219.  Force  of  a  BIotv  In  order  to  express  the 
.inioiinl  of  force  between  the  face  of  a  hammer,  for  in- 
sliiiur,  and  the  head  of  a  nail,  wc  must  consider  what 
Wfijiht  ni'ist  he  laid  upon  tiie  iiead  of  the  nail  to  force  it 
into  the  wood.  A  nail  recpiires  a  larjji'  force  to  pull  it  out, 
when  friction  alone  is  retaining  it,  and  to  force  it  in  must 
of  course  reipiire  a  still  larger  force. 

Xow  the  head  of  the  hammer,  when  it  delivers  a  Idow 
upon  the  head  of  the  nail,  must  be  capable  of  develo})ing  a 
force  equal  for  a  short  time  to  the  continued  pressure  that 
would  be  produced  by  a  very  heavy  load.  Hence,  the  effect 
of  the  hanuner  may  be  explained  by  the  principles  of  enciyi/. 
When  the  hammer  is  in  motion  it  lia.5  a  <[uantity  of  kinetic 
energv  stored  up  in  it.  and  when  it  comes  in  contact  with 
the  nail  this  energy  is  instantly  converted  into  work  which 
forces  the  nail  into  the  wood. 

EXAMPLES. 

1.  Suppose  that  a  hammer  weighs  1  lb.,  and  that  it  is 
impelled  downwards  by  the  arm  with  considerable  force,  so 
that,  at  the  instant  the  head  of  the  hammer  reaches  the 
niiil.  it  is  moving  with  a  velocity  of  20  ft.  ))er  second  ;  find 
tiie  force  which  the  hammer  exerts  on  the  nail  if  it  is 
driven  into  the  wood  one-tenth  of  an  inch. 

Let  P  be  the  force  which  the  hammer  exerts  on  the  nail, 
tiien  the  work  done  in  forcing  the  nail  into  the  wood  = 
/'  X  j^Ts,  and  the  energy  stored  up  in  the  hammer 


—  Iniri 


=  i»l 


(14 


fi. 


Siiu'c  the  work  doiu'  in  forcing  the  nail  into  the  wood 
niusi  be  e(puil  to  all  the  work  stored  in  the  hammer,  (Art. 
•.M ',),  we  have 


r^o 


-  =  (>:i ; 


/'  =  744  lbs. 


413 


EXAMPLES. 


Ilenee  the  force  which  tlio  lianimor  exerts  on  the  head  of 
the  null  is  at  least  Ti-l  lbs. 

2.  If  the  hanimcr  in  tiie  last  example  forces  the  nail  into 
the  wood  only  O.Ul  of  an  int'li,  lind  tiie  force  exerted  on 
the  nail.  Ahx.   T44<)  lbs. 

Hence,  we  see  tliiit,  iicpor<liii<r  us  the  wood  is  harder,  /.  ,".,  arcord 
ing  as  the  nail  enters  h'ss  at  each  stroke,  the  forct;  of  tlie  l)lo\v 
iH'Coines  jjreater.  So  that  wlien  we  speak  of  the  "  force  of  a  blow," 
we  must  spirify  how  soon  the  body  giving  the  IjIow  will  come  to 
rest,  otherwise  the  term  is  nieuniiiglese.  Tims,  sui-pose  a  hall  of 
100  lbs.  weight  have  ii  velocity  that  will  cnus);  it  to  ascend  1000  ft.; 
if  l'  e  ball  ib  to  be  8to|.])ed  at  the  end  of  1000  ft.,  a  force  of  100  lbs. 
will  do  it ;  but  if  it  is  to  be  stoj)ped  at  the  end  of  one  foot,  it  will 
need  a  force  of  100000  lbs.  to  do  it ;  and  to  stop  it  at  the  end  of  one 
inch  will  require  a  force  of  '200000  lbs.,  and  so  on. 


220.  "Work  of  a  Water-Fall.— When  water  or  any 
Ixtdy  falls  from  a  <fiven  heifflit,  it  is  plain  that  the  work 
which  is  stored  up  in  it,  and  which  it  is  cai)a"l)le  of  doing,  is 
ccpial  to  that  wiiieh  wonld  be  re(|nired  to  raise  it  to  the 
hci<,dit  from  which  it  has  fallen  ;  /.  v.,  if  1  lb.  of  water 
de.-^cend  ih rough  1  foot  it  must  accumulate  as  much  work 
as  wonld  be  required  to  rai.se  it  throngii  1  foot.  Ilenee 
when  ii  fall  of  water  is  cmplo^-ed  to  drive  a  water-whed,  or 
any  other  hydfiuilic  machine,  whose  modulus  is  given,  the 
work  done  ujxm  tlic  mtichine  is  eipuil  to  the  weight  of  the 
water  in  pounds  x  its  fall  in  feet  x  the  modulus  of  the 
machine. 

EXAMPLES. 

1.  The  breadth  of  a  .-stream  is  /;  feet,  depth  a  feet,  mean 
velocity  /•  feet  per  uiiiiufe.  mid  the  h.'ight  of  tiie  fall  //  feet; 
lind  (1)  the  horse-pnucf.  .\'.  of  the  water-wheel  who,se 
modidus  is  ///.  and  {-i)  lind  I  he  number  of  cubic  feet,  A, 
which  the  wheel  will  pump  per  minute  from  the  bottom  of 
the  fall  to  the  height  of  //,  feet. 


y;.v.i.i/;  /,i:s. 


ii;} 


on  the  head  of 


L'os  tlif  nail  into 
oreo  exerted  on 
Ills.   7440  11)8. 

anliT,  (.  e.,  accord 
f()rc(!  of  tlie  blow 
"  force  of  II  blow," 
blow  will  come  to 
Hiiiiposc  a  ball  of 
to  ascend  1000  ft. ; 
I  force  of  100  lbs. 
of  one  foot,  it  will 
at  the  end  of  one 


water  or  any 
1  that  the  work 
a"l)lc  of  doing,  is 
<  raise  it  to  the 

1  11).  of  water 
(■  as  mnch  work 
1  foot.  Hence 
water-wheel,  or 
IS  is  given,  the 
p  weight  of  the 
modulus  of  the 


h  n  feet,  mean 
r  the  fall  //  feet ; 
cr-wlieel  wlio,se 
f  cuhic  feet,  A, 
I  the  bottom  of 


Weight  of  water  going  over  tlif  fall  per  niin.  =  Cyllt  aor. 
.-.     Work  of  wheel  jicr  niin.  =  I'ri.'t  tifu'/im.  (1) 


O'.'.,')  iibrlini 


(■i) 


Work  in  pumping  water  per  niin.  =  02.5  .1//,; 

whieii   must  =  the   work  of   the   wheel    i)er  niin.;    hence 
from  (1)  we  have 

G2.5  J/.i  =  ()2..")  a/tr/im; 


(ibi'/tni 
A  =  —,—  • 


(3) 


2.  The  mean  section  of  a  stream  is  5  ft.  by  2ft.;  its 
mean  velocity  is  35  ft.  per  iniiiut(! ;  there  is  a  fall  of  1.3  ft. 
on  tliis  stream,  at  which  is  erected  a  water-wheel  whose 
woduhis  is  0.C5  ;  find  the  horse-power  of  tiie  wheel. 

Ans.  5.0  1 1. -I'. 

3.  In  how  many  hours  would  the  wheel  in  Ex.  2  grind 
8000  bushels  of  wheat,  supposing  each  horse-power  to  grind 
1  bushel  per  hour?  Auh.   14284  hours. 

4.  llow  many  cuhic  feet  of  water  must  be  made  to 
descend  the  fall  per  minute  in  Kx.  2.  that  the  wheel  may 
grind  at  the  rate  of  28  bushels  per  hour? 

Am.   1750  cu.  ft. 

5.  Given  the  stream  in  Ex.  2,  what  must  be  the  height 
of  the  fall  to  grind  10  bushels  jier  hour,  if  the  modulus  of 


the  wheel  is  0.4  'i 


.l«,s.  3r.r  feet. 


0.  Find  the  useful  horse-power  of  a  water  wheel,  sup- 
])osing  the  stream  to  be  5  ft.  broad  and  2  ft.  deep,  and  to 
flow  witli  a  velocity  of  30  ft.  ]H'r  minute:  the  height  of  the 
fall  being  14  ft.,  and  tiie  modulus  of  tiie  machine  0.05. 

.Im.s.   5.2  nearV. 


"T 


414 


EXAM1'I,KS. 


221.  The  Duty  of  an  Engine.—  The  (hifi/  of  an  pixjinf 
/.v  llic  number  of  units  of  work  irhicli  it  is  cdpahlc  of  (/di/ii/ 
by  burniiKj  a  ffircii  t/nan/ifi/  if  /'nrh — ft  Iiiis  liocii  IoiiihI  1)\ 
I'xiH'riment  that,  wliattner  may  \)v  the  iirossiirc  at  wliicli 
tlic  steam  is  formed,  the  (itiantiry  of  fuel  iieeessarv  tu 
eva|)<)ratc  a  given  volume  of  water  is  always  nearly  the 
same;  hence  it  is  most  advantageous  to  employ  steam  of  a 
high  pressure.* 

Ill  good  ordinary  pngiiics  the  duty  varies  between  200000  and 
500000  units  of  work  for  a  lb.  of  coal.  The  extent  to  wliich  tlie 
economy  of  fuel  niiiy  be  earried  in  well  illustrated  by  the  engiuet*  em- 
ployed to  drain  the  mines  in  ("ornwall,  England.  In  1815,  the 
average  duty  of  these  engines  was  20  million  units  of  work  for  a 
bushelf  of  coal ;  iu  1843,  by  reason  of  giiec-ssive  improvements,  the 
HTcrage  duty  had  become  00  millions,  ert'ecting  a  saving  of  £85000 
])er  annum.  It  is  stated  that  in  the  case  of  one  engine,  the  duty  was 
raised  to  l«o  millions.  The  duty  of  the  engin(>  depends  largely  on 
the  construction  of  the  boiler;  1  lb.  of  coal  in  the  Cornish  boiler 
evaporates  11?.  lbs.  of  water,  while  iu  a  ditferently-shaped  boiler  8.7 
is  the  maximum.^ 

E;X  A  M  P  I,  ES  , 

1.  An  engine  burns  2  lbs.  of  coal  for  each  horse-jjower 
per  hour  ;  find  the  duty  of  the  engine  for  ii  lb.  of  coal. 

ilere  the  work  dojie  in  one  hour 

=  60  X  33000  foot-pounds; 

therefore  tiie  duty  of  the  engine  ^  30  x  33000  foot-pounds, 

—  OilOdOO  foot-pounds. 

2.  IIow  many  bushels  of  coid  must  be  expended  in  a 
day  of  ^*4  hours  in  raising  150  cubic  ft.  of  water  per  minute 

♦  8i!e  Tiile  In  MoclmiilcH'  Mnjrazliic,  hi  tlio  vcnr  1R41. 

+  Oue  liusliol  of  coal  =  HI  or  M  Ills.,  (lc|icmliiii;  upon  ulicro  il   is.     (iooilcvr, 

X  Hourni!  on  Ihu  SU'aui  Enginf,  p.  171,  iiiid  Kar'miin,  Unrful  Inforniiulou, 
,1    177 


jia 


T 


/  nf  on  riifji/if 
Hthli'  I  if  ( III  i  11(1 
ocn  IoiiihI  1)\ 
lire  iit  wliicli 
iit'L'ossarv  tu 
,s  nearly  the 
y  steam  of  a 


en  200000  iiiui 
It  to  wliicli  the 
the  engines  eni- 
In  1815,  the 
1  of  work  for  a 
)rovemontH,  the 
.•ing  of  £85000 
p,  the  duty  was 
^nds  largely  on 
Cornish  holler 
laped  boiler  8.7 


horse-))o\ver 
of  coal. 


foot-pounds, 
it-poiiiuls. 

pended   in  ii 
'V  per  minute 

■«  il   is.     OcHxIcvi', 
('fill   Infiiriiiiitiiiu, 


WOUH    OF  A     VAinAHLl-:   F()l{CE. 


415 


from   a  dei)tli   of   100  fathoms  ;    tlie  duty  of   the   engine 
licing  (JO  millions  for  a  bushel  of  coal  ? 

J//,v.   135   hiishels. 

.'5.  A  steam  engine  is  required  to  raise  TO  cuhic  ft.  of 
water  jier  minute  from  a  depth  of  800  ft. ;  liiul  how  many 
Ions  of  eoal  will  be  reijuired  per  day  of  24  hoin's,  supposing 
the  duty  of  the  engine  tobo  250000  for  a  lb.  of  coal. 

Am.  ti  tons. 


222.  Work  of  a  Variable  Force.— When  the  force 
which  performs  work  through  a  given  space  varies,  the 
work  done  may  be  determined  by  multiplying  the  given 
si)acc  by  the  mean  of  all  the  variable  forces. 

Let  AG  represent  the  space  in  units 
of  feet  through  whieh  a  variable 
force  is  exerted.  Divide  AG  into 
six  equal  parts,  and  su])pose  /',,  /'g, 
I\,  et".,  to  be  the  forces  in  pounds 
ui)i)lied  at  the  points  A,  B,  C,  etc., 
nspeetively.  At  these  ])oiuts  draw  the  ordinates  1/1,1/3,  y^, 
etc.,  tc  re])rescnt  the  forces  which  act  at  tlie  })oints  A,  B, 
0,  etc.  Then  the  work  done  from  A  to  B  will  l)e  equal  to 
the  space,  AB,  multiplied  by  the  mean  of  the  forces  1\ 
and  /',,  i.e.,  the  work  will  be  re])resentcd  by  the  area  of 
the  surface  Aal/K  In  like  manner  the  work  done  from 
B  to  0  will  be  represented  by  the  area  \\bcC,  and  so 
on  ;  so  that  the  work  done  through  the  whole  space,  AG, 
by  a  force  which  varies  continuously,  will  be  represented  by 
the  area  Ar/r/G.  This  area  may  be  found  approximately  by 
(he  ordinary  rule  of  .l/r«,sv'm//f/»  for  the  area  of  a  curved 
surface  Avith  e((uidislant  ordinates,  or  more  accurately  by 


B    c 


Fig.  09 


Simpson's*  rule,  the  proof  of  which  we  shall  now  give. 
223.  Simpson's   Rule.  -Tict   //,,  //j.  ?/,,  etc.,  bo   the 

♦  Alili(Hij,'li  II  wii'-  not  liivi     I'd  by  SimpHon,    Sou  Toilhuiitch 


i^m 


410 


SI.UI-.SD.VS    /.TAJ?. 


e(|iiitlistiiiit  ordiiiatcH  (Fifj.  Si))  and  /  tlso  distam'o  l)etweon 
aiiv  two  coiisecutive' ordiiiuU's ;  tlioii  liv  taking  tlic  snni  of 
tlif  (rapczoids,  AabH,  liOrC,  etc..  we  liave  Uiv  the  area  (if 
A(i(/(J, 

jMz/i  +  ^2)  +  yo/i  +  Hi)  + 1'(//3  +  ux)  +  i^'c- 

=  Vd/i  +  -^i/s  +  •-V/3  +  -!/i  +  -^1/,  +  -'z/6  +  //i);  (0 

winch  i.-i  tlio  di'dinary  formula  of  uuMisiiration. 

Now  it  is  ovidont  that  wlioii  the  curve  is  always  concave 
to  the  line  ACJ  (I)  will  give  i<»»  .undl  a  result,  and  ii'  con- 
vex it  will  give  loo  large  a  resuli. 

Let  Fig.  1)0  rei»re.>jenl  the  sjjace  hetwecii  any  two  odd 
consecutive  oi'dinates,  as  (V;  and  Ke(Fig.  8!») ;  divide  CE 
into  tln-ce  ei|ual  i)ii.rt,«,  CK  —  KL  =  LK,  ^  ^_  rf  i 
and  erect  the  ordinates  Kk  and  L/,  dividing 
the  two  trapezoids  CVv/D  and  IWt-E  into  the 
three  trapezoids  (V/'K,  K/('/L,  and  IJrK.-  ^ — k~pl" 
The  sum  of  the  aretis  of  t hese  throe  trapezoids  Fig.M 

=  ^C;K  ((V;  -f  2Kk  +  21./  +  Kc) 

=  y  {Ce  -f  -iKl-  +  2T./  +  E^).  (since  \VK  =  iOD  -^  j/) 

=  |/(Cf;  +  Wo  -f  Er).  (since  '^K^-  +  2\J  =  4I)o).        (2) 

which  is  a  closer  a])proxinnition  lor  the  area  of  VerK 
than  (1). 

Now  when  the  cnrve  is  concave  towards  CE.  (2)  is 
smaller  than  the  area  lietween  CE  and  the  curve  r/v/A';  if 
we  substitute  for  \)ii,  the  ordiiuite  IW,  which  is  a  little 
greatei'  than  Do  and  wiiicii  is  given,  (2)  hecomes 


which  is  a  still  closer  approximatiiu  tlian  (2). 


(3) 


h:XAMI'LhS. 


41 : 


(:5) 


stii))c'i'  between 
iiig  tlic  sum  of 
for  I  lie  iiivii  of 


in. 

always  conotive 
lit,  and  ir  con- 

1  any  two  oild 
<!t);"(liviae  CK 

— LLI I 

C        K  D  L       E 
8  Fig.90 


=  U^n  -.  io 

==  4I)o),       i'i) 
area  of  (JaK 

pds  CE.  (2)  is 
curve  ck<Ur\  if 
iiieli  is  ii  Hi  lie 
>me8 


Similarly  we  liave  for  the  areas  of  AflfK'  and  Ef^CJ. 

y.  {\,i  +  4!V>  4-  Ct),  and  V  (l'>  +  -^^Y  +  <V/)-      C-^) 

Adding  (:3)  and  (4)  logether,  we  have   for  an  approximate 
value  of  the  whole  area, 

w/;/r//  /.v  Siinpaoii's  Fonimld.  Hence  Simpson's  rule  for 
findinsr  the  area  approximately  is  tiie  following:  Divide  the 
nhscis.sd,  AO,  in/o  an  eirii  vumhvr  n/ri/nnl  jxir/s,  mid  erect 
ordinuten  at  the  puin/s  of  dirixion  ;  t/icii  adil  lo;/etIicr  the 
first  and  last  ordinatps,  twice  tJie  sani  of  all  the  other  odd 
jrdiuates,  andfoar  times  the  sum  of  all  the  even  ordinate^  ; 
multiply  the  sum  Inj  one-third  of  the  common  distance 
hetween  any  two  adjareut  ordinatrs.  (See  'I'odhnnter's 
Mensuration,  also  Tale's  (ieonietry  and  Mensuration,  also 
Morin's  Moch's,  by  Bennett.) 

EXAMPLES. 

1.  A  variable  force  has  act'>d  throngli  .T  ft.;  the  value  of 
the  force  taken  at  seven  successive  e(iui.)istanl  points, 
including  the  first  and  the  last,  is  in  ll)s.  IhO,  If)!. 2,  1^*', 
108,  (14. T),  84.  75. f!  ;  lind  the  whole  work  done 

.l//,v.   :i4(!.4  foot-pounds. 

•i.  A  variable  force  has  acted  through  (1  ft.;  the  value  of 
the  force  taken  at  seven  successive  e(|uidistant  ])oinls, 
including  tho  first  am'  the  last,  is  in  lbs.  3,  8,  15,  24,  :ib, 
48,  G;}  ;  find  tho  whole  work  done. 

.1  //.•.'.   I<t2  foot-pounds. 

3.  A  varialile  force  has  acted  Ihrougli  Si  ft.;  the  value  of 
the  force  taken  at  seven  successive  eiiuidistant  points, 
iiu'luding  the  first  and  the  last,  is  in  lbs.  (1.082,  (!.l()4, 
(;,24r>,  (>.:{2r>,  tUO.J,  (1.481,  (;..5r)T;   find  the  wliole  work  done. 

.|/;.v.   .")(i.'.t07  foot-pounds. 


^m 


418 


^:xA^rph^:s. 


SlionkI  any  of  tlio  ordiiiiili's  lu'cunu'  zito.  it  will  not  pro- 
vcnl  (lie  nsc  of  Simpson's  nilo. 

Simpson's  imiIc  is  iippliciiliic  to  otiici'  invest igiit ions  as 
well  as  to  (liat.  of  work  done  by  a  varial)l('  force.  For 
(■.\ainj)Ie,  if  we  want  tlie  velocity  fjenerated  in  a  given  time 
ill  a  particle  liy  a  varialtle  force,  let  the  straight  line  A(J 
represent  the  wiiolc  time  (hiring  which  the  force  acts,  anil 
let  the  straight- lines  at  right  angles  to  AG  represent  the 
force  at  the  corresponding  instants;  tiien  the  area  will 
represent  the  whole  space  descrilied  in  the  given  time. 


E  :.  A  M  P  L  E  s  . 

1.  The  ram  of  a  jiile-driving  engine  weighs  half  a  ton,* 
and  lias  a  fall  of  17  ft.;  how  many  niiits  of  work  are  |)er- 
formed  in  raising  this  ram  ?        Aiis.    1!)()4()  foot-pounds. 

"i.  How  many  units  of  work  are  rc(Hiired  to  raise  7  cwt 
of  coal  from  a  mine  whose  depth  is  i;$  fathoms  ? 

A  US.   (11152  foot-pounds. 

',].  A  horse  is  used  to  lift  the  earth  from  a  trench,  which 
he  does  by  means  of  a  cord  going  over  a  pulley,  lie  pulls 
up,  twice  every  r>  minutes,  a  miui  weighing  \'M)  lbs.,  and  a 
barrowful  of  earth  weighing  ::i<jO  Ills.  Kach  time  the  horse 
goes  forward  40  ft.;  find  the  units  of  work  done  by  the 
horse  iier  hour.  Ans.  3T-t4()(). 

4.  A  niilwiiy  train  of  7'  tons  ascends  an  inclined  jilanc 
which  has  a  rise  of  c  ft.  in  100  ft.,  with  a  uniform  speed  of 
))i  miles  per  liour ;  find  the  horse-power  of  the  engine,  the 
friction  being  j:;  lbs.  per  ton. 

mT{p  +  22Ae) 

5.  A  railway  train  of  80  tons  ascends  an  incline  which 
rises  one  foot  in  Soft.,  with  the  uniform  rate  of  1.")  miles 


•  Onr  1(111       aicvM.  -    a-.Mlllbu. 


i:.\A}fi-Li:s. 


41!) 


t  will  not  pre- 

vosiigiilioiis  as 
•lo  fiircc.  For 
II  ii  given  time 
liglit  liiu'  A(i 
force  acts,  and 
represent  (lie 
the  area  will 
ven  time. 


lis  half  a  ton,* 
«'ork  are  j)er- 
foot-pounds. 

I)  raise  7  cwt. 

s? 

oot-pounds. 

trench,  wliieh 
ey.  lie  pulls 
30  lbs.,  and  a 
ime  the  liorso 
done  hy  the 
»s.  3:4400. 

iicliiied  plane 
orni  speed  of 
c  engine,  the 

---^  H.-P. 

incline  which 
['  of  1")  miles 


Alls,   w  = 


per  hour  ;  find  the  horse-jxjwer  of  the  engine,  the  friction 
being  S  lbs.  per  ton.  Aiis.   l(;s.9(J  H.-P. 

C.   If  a  horsf  exert    a  traction  of  /  lbs.,  what  weight, /r. 

will   he  pull   up  or  down  a  hill  of  small  inclination  which 

has  .1  rise  of  c  in  100,  the  coetficient  being  /"? 

100/ 

100/ ±"«' 

T.  From  what  depth  will  an  engine  of  22  horse-power 
raise  13  tons  of  coal  in  an  hour?  .l//v.   U'.Mi  ft. 

8.  An  engine  is  observed  to  raise  7  tons  of  material  an 
hour  from  a  mine  wlu'se  dejith  is  S.l  fathoms ;  find  the 
hor^e-i)ower  of  the  engine,  supposing  ^  of  its  woik  to  be 
lost  in  transmission.  Ann.   4.S4(i.5  II.-P. 

il.  Re(|uired  the  horse-power  of  an  engine  that  would 
supply  a  city  with  water,  working  12  hours  a  duy,  the 
water  to  be  raised  to  a  height  of  50  ft.;  the  number  of 
inhabitants  b^Iug  i.JOOOO,  and  each  person  to  use  5  gallons 
of  water  a  day,  the  gallon  weighing  8^  lbs.  nearly. 

Alls.   11.4  H.-P. 

10.  From  what  de])th  will  an  engine  of  20  liorse-powcr 
raise  000  cubic  feet  of  water  per  hour  ?      .Iw.v.   1050  feet. 

11  At  wliat  rate  per  hour  will  an  engine  of  30  horse- 
jiower  draw  a  train  weighing  00  tons  gross,  the  resistance 
being  8  lbs.  per  ton  ?  Ans.   15.G25  miles. 

12.  What  is  H'.c-  gro.-s  weight  of  a  train  which  an  engine 
jf  25  horse-power  will  draw  at  the  rate  of  25  miles  an 
hour,  resistances  being  8  lbs.  per  ton? 

An^.  40.875  tons. 

13.  A  train  whose  gross  weight  is  80  tons  travels  at  the 
rate  of  20  miles  an  hour ;  if  the  resistance  is  8  lbs. 
per  ton,  what  is  the  horse-power  of  the  engine  ? 

Am.  34,1,  H.-P. 


420 


EXAMPLES. 


14.  What  must  bo  tlio  l('ii<i;th  of  the  stroke  of  ii  piston 
of  an  engine,  the  siirf'aee  of  wliieh  is  1500  B<iuaro  inches, 
which  makes  '10  strokes  per  minute,  so  tliat.  with  a  mean 
pre.vsure  of  12  lbs,  on  eaci)  scpuire  inch  of  the  piston,  the 
engine  may  be  of  SO  horsc-jtower  ?  Anx.   'i\  ft. 

15.  The  diameter  of  the  piston  of  an  engine  is  SO  ins., 
tile  leugtli  of  tile  stroke  is  10  ft.,  it  makes  11  strokes  ])er 
minute,  and  the  mean  pressure  of  tlie  steam  on  the  piston 
is  12  lbs.  per  square  inch  :  wiial  is  the  horse-power  ? 

Anx.  201 -Ofj  1 1. -P. 

IG.  The  cylinder  of  a  steam  engine  has  an  internal 
diameter  of  3  ft.,  the  length  of  the  stroke  is  O  ft.,  it  makes 
(i  strokes  jier  minute:  under  what  effective  pressure  per 
square  inch  would  it  have  to  work  in  order  that  75  horse- 
power may  be  done  on  the  piston  ?  Anx.  G7-  54  lbs. 

17.  It  is  said  that  a  horse,  walking  at  the  rate  of  2^  miles 
an  iiour,  can  do  1050000  units  of  work  in  an  hour  ;  wliat 
lorce  in  pounds  does  he  continually  exert  ? 

Anx.    125  lbs. 

18.  Find  the  horse-jjower  of  an  engine  which  is  to  move 
at  the  rate  of  150  miles  an  hour,  the  weight  of  tiic  engine 
and  load  l)eiiig  50  tons,  and  the  resistance  from  friction 
U;  lbs.  per  ton.  Ans.    04  li.-P. 

1!).  There  were  0000  cubic  ft.  of  water  in  a  mine  whose 
depth  is  00  fathoms,  when  an  engine  of  50  horse-power 
began  to  work  the  pump  ;  the  engine  continued  to  work  5 
hours  before  tiie  mine  was  cleared  of  the  water  ;  required 
the  number  of  cubic  ft.  of  water  wiiieh  had  run  into  the 
mine  during  the  5  hours,  supposing  |  of  the  work  of  the 
engine  to  be  lost  by  transmission.       Ans.  10500  cubic  ft. 

20.  Find  the  horse-power  of  a  steam  engine  which  will 
raise  30  cubic  ft.  of  water  per  minute  from  a  mine  440  ft. 
<leep.  Ans.  25  ll.-l'. 


ike  of  ii  ])iston 

I  8(Hiare  inches, 

1     with    11    UK'tlM 

the  pi,st(iii,  tile 
A)is.  ;^  ft. 

gine  is  SO  ins.. 

I I  strokes  per 
on  the  piston 

1)0 wer  ? 
!(!l-0(j  II. -P. 

IS  an  internal 
i)  ft,,  it  makes 
13  pressure  per 
•  that  75  horse- 
?.  or -54  lbs. 

rate  of  'il  miles 
iin  hour  ;  what 

US.    laS  lbs. 

ich   is  to  move 
of  the  engine 
;  from  IVietion 
ns.   04  II.-P. 

1  a  mine  whose 
)0  iiorse-power 
uied  to  work  5 
ater  ;  re(|nireil 
1  run  into  the 
he  work  of  the 
500  cubic  ft. 

ino  which  will 
a  mine  440  ft. 
ns.  J*5  li.-I'. 


EXAMPLES. 


421 


21.   If  a  pit   10  ft.  deep  with  an  area  of  4  s(iuare  ft.  i)e 
OAsavated  and  the  earth  thrown  up,  how  mncli   work  will 
have  been  done,  supposin-x  a  cubic  foot  of  earth  to  weigh 
■i  .)o  lbs.  Alls.  18000  ft.-lbs. 

I  22.   A  well-shaft  :{0<)  il.  deep  and  5  ft.  in  diameter  is  full 

of  water;  iiow  many  units  of  work  must  l)e  expended  in 
-'ettinsr  this  water  out  the  well  ;  (/.  e.,  irrespectively  of  any 
other  water  Howiug  in)?  I "•>•■•   r)522:J2()2  ft.-lbs. 

215.   A  shaft  a  ft.  deep  is  full  of  water;  lind  the  depth  of 
the   surface   of   the   water   when  one-(iuarter  of   the  work 


recpxired  to  empty  the  shaft  has  been  done. 


Atis.  "ft. 


24.  The  diameter  of  the  cylinder  of  an  engine  is  80  ins., 
the  piston  makes  per  minute  S  strokes  of  10^  ft.  under  a 
mean  pressure  of  15  li)s.  per  sipuire  inch  ;  the  modulus  of 
the  engine  is  0-55;  how  nuiuy  cubic  ft.  of  water  will  it 
raise  from  a  depth  of  112  ft.  in  one  minute? 

Alls.  485- 78  cub.  ft. 

'  25.   If  in  the  last  example  the  engine  raised  a  weight  of 

r)(i4:3:5  lbs.  through  tlO  ft.  in  (me  minute,  what  must  be  the 
mean  pressure  per  s(iuare  inch  on  the  piston  ? 

Ans.  26-37  lbs. 

2().  If  the  diameter  of  the  piston  of  the  engine  in  Ex.  24 
had  been  85  ins.,  what  addition  iu  horse-power  woiild  that 
nuike  to  the  useful  power  of  the  engine  ? 

Alls.  13-28  II.-P. 

27.  If  an  engine  of  50  horse-power  raise  28(1(1  cub.  ft.  of 
water  per  hour  from  a  mine  00  fathoms  deep,  liml  the 
modulus  of  the  engine.  Aiis.  -05. 

28.  Find  at  what  rate  an  engine  of  30  ho'rse-powcr  could 
draw  a  train  weigiiing  50  Ions  up  an  incline  of  1  in  280, 
Ihv  resistance  from  friction  being  7  il)s.  per  Ion. 

,l//.v.   1320  ft.  per  minute, 


ita 


45J2 


KXA.VI'LKS. 


'V.K  A  forgo  Iiiimmer  wi-l^liing  ;}()()  lli.s.  niakos  100  lifts  a 
minute,  tlio  porpi'iKiicular  iici;,f|it  <>f  cacli  liCl  \mu^  2  ft.; 
what  is  llic  horso-powor  uf  tlio  I'ligiiio  that  gives  isiDtioii  to 
^0  such  ham iiiers?  Jh.v.  ;!(;.;5(;  n..l>. 

.'iO.  An  "ugiiu!  of  ].)  horse-power  raises  -iitOO  :l)s.  of  eoal 
from  a  pit  r^OO  ft.  deep  in  an  liour,  and  also  gives  motion 
to  a  hammer  which  mukes  50  lifts  in  a  minute,  eaeli  lift 
having  a  i)eri)en(licular  height  of  4  ft.;  what  is  the  w.'ight 
of  the  hammer  ?  Ann.   1250  lbs. 

'.U.  Find  the  liorsr-power  of  i!ie  engine  to  raise  7' tons  of 
coal  i)er  hour  from  a  [lit  whose  depth  is  a  fathoms,  and  at 
the  same  time  to  give  motion  to  a  foim-  hammer  weiirhiii<>- 
w  lbs.,  which  makes  n  lifts  per  minute  of  h  ft.  each. 

3-3000 

32.  Find  the  useful  work  done  by  a  fire  engine  per 
second  whicli  discharges  every  second  13  lbs.  of  water  with 
a  velocity  of  50  ft.  \m'  second.  Aiim.  508  nearly. 

33.  A  railway  truck  weighs  m  tons  ;  a  horse  draws  it 
along  horiz(mtally.  the  resistance  being  n  lbs.  jter  ton  ;  in 
passing  over  a  space  .»■  the  velocity  clianges  from  u  tor; 
find  tho  work  done  by  the  liorse  in  this  space. 


A  us. 


(' 


i(')  +  ni/ts. 


34.  The  weight  ^f  a  ram  is  GOO  lbs.,  and  at  the  end  of 
the  blow  has  a  velocity  of  32^  ft.;  what  work  has  been 
done  in  raisin"  it  ?  J;(,v.   '.xi.'io. 

'.)').  l?t'.'jUired  th"  work  stored  in  a  cannon  bal!  whose 
weight  is  '.i-^  lbs.,  and  velueily  15(10  ft.        Aiis.    H25(JOO. 

3fi  A  ball,  weighing  -JO  lbs..  i>  |irojected  with  a  velocity 
of  tic  ft.  a  second,  on  a  liowlin'r-greeii  ;  what  space  will  the 
ball  move  over  bebire  it  comes  to  rest,  allowing  llr.'  IVietion 
to  be  ^'f^  the  weight  of  the  ball?  .J/(.v.    loor-3  ft. 


mmm 


EXAMPLES. 


433 


;c'.s  !(»(»  lifts  a 
t  hviuyr  :>  n.; 
c'S  motioii  t(» 

iii-;5(;  ii.-p. 

0  il)s.  of  coal 
frivi'.s  morion 
into,  oiU'li  lift 
is  the  w.'iglit 
.  1250  lbs. 

ii.se  T  tons  of 
loais,  ami  at 
nor  weighing- 
each. 
nhir 


\\.-\\ 


J  engine  per 
•f  water  with 
J08  nearly. 

rse   draws   it 

per  ton  ;  in 

rom  H  to  r  ; 

t^)  +  inns. 

\  the  end  of 
rk  lias  hi.'t'n 
\us.  '.)(;.■>(). 

liali    whose 

iri:)0()(». 

til  a  vdoeil y 
laee  will    I  lie 
!lr.'  IVictinii 
l(l(ir';{  ft. 


:J7.  A  train,  weighing  19:J  tons,  has  a  velocity  of  ;U) 
miles  an  hour  when  the  .steam  is  tu.'ned  off;  how  far  will 
liie  train  move  on  a  level  rail  before  coming  to  rest,  the 
friction  being  .")i  !b.s.  })er  ton  y  Ann.   12250  ft. 

;{8.  A  train,  weighing  00  Ioils.  has  a  velocity  of  40  miles 
an  hour,  when  the  steam  is  turned  off,  how  far  will  if 
ascend  an  inchne  of  1  in  100,  taking  friction  at  8  lbs.  a  ton  ? 

A  us.  3942^  ft. 

;51».  A  carriage  of  1  ton  moves  on  a  level  rail  witfi  the 
speed  of  H  ft.  a  second ;  through  what  s{)ace  must  the 
carriage  move  to  have  a  velocity  of  2  ft.,  sujjposing  friction 
to  he  0  lbs.  a  ton?  Ans.  348  ft. 

40.  If  the  carriage  in  the  last  example  moved  over  400 
feet  before  it  comes  to  a  state  of  rest,  what  is  the  resistance 
of  friction  per  ton?  Afis.  5.5T  lbs. 

41.  A  force,  P,  acts  upon  a  body  parallel  to  the  plane; 
liiid  the  space,  .v,  moved  over  when  the  body  has  attained  a 
given  velocity,  /',  f  lie  coe(li(nent  of  friction  being  /',  and  the 
hodv  weighing  w  lbs.  ,  7vr^ 

4'3.  Suppose  the  body  in  the  last  example  to  be  moved 
for  /  seconds;  rc([iiired  (1)  the  velocity,  r,  acrpiired,  and 
(2)  the  work  stored. 


An.s.  (1) 


/' 


\'^.  A  bo(^y,  weighing  40  lbs.,  is  projected  along  a  rough 
Imrizontal  ])!ane  with  a  velocity  of  I50  ft.  per  second  ;  the 
coetlicient  of  friction  i-  \\  lind  the  work  done  against  fric- 
li'in  in  ")  seconds.  Anx.  3500  foot-pounds. 

44.  .\  body  weighing  50  lbs.,  is  projected  along  a  rough 
horiziiiital  iilaiic  with  the  velocity  of  40  yards  jicr  second  ; 
'Mill  the  work  expended  when  the  body  comes  to  rest. 

' Ans.  11250  ft.-lbs. 


r 


424 


EXAMl'LES. 


45.  If  a  trail!  of  cars  weigliing  100000  lbs.  is  moving  ol 
a  horizontal  track  with  a  velocity  of  40  miles  an  hour  when 
the  steam  is  turned  off;  through  wiiat  space  will  it  move 
before  it  is  brought  to  rest  by  friction,  tiie  friction  'leing 
8  lbs.  per  ton  P  A  as.   133T4.8  ft. 

4G.  V/hat  amount  of  energy  is  acquired  by  a  body  weigh- 
ing 30  lbs.  that  falls  through  the  whole  lengtli  of  a  rougli 
inclined  plane,  the  height  of  which  is  30  ft.,  and  the  base 
100  ft.,  the  coefficient  of  friction  being  \  ? 

Ans.  300ft.-lbs. 

47.  If  a  train  of  cars,  weighing  y  tons,  ascend  an  incline 
having  a  raise  of  e  ft.  in  100  ft.,  with  the  velocity  i\  ft.  per 
second  when  the  steam  is  turned  off;  through  what  space, 
X,  will  it  move  before  it  comes  to  a  state  of  rest,  the  friction 
being  jo  lbs.  per  ton  ?  ,««  .    ^  _        11~0('„2 

^i^Ac  +p)' 


Ans.  X 


48.  Suppose  the  train,  in  Ex.  4,  Art.  217,  to  be  attached 
to  a  rope,  passing  round  a  wheel  at  the  top  of  the  incline, 
which  has  an  empty  train  of  1\  tons  attached  to  the  other 
extremity  of  the  rope:  what  velocity,  r,  will  the  traia 
acquire  in  descending  *■  ft.  of  the  incline  ? 


/        T  —  "i 
Ans.   V  =  Y  V' y-qr^ 


'7\ 


+  T, 


gps_ 
1120' 


40.  An  engine  of  35  horse-power  makes  20  revolutions 
per  minute,  the  weight  of  the  fly-wheel  is  20  tons  and  the 
diameter  is  20  ft.;  what  is  the  accumulated  energy  in  foot- 
pounds? Ans.  307054. 

50.  If  the  fly-wheel  in  the  last  example  lifted  a  weight  of 
4tM)()  |l)s.  tlirougii  3  It.,  wiiat  [»art  of  lis  angular  velocity 
would  it  lose  ?  Ans.  gif. 

51.  If  the  axis  of  the  aliove  fly-wheel  be  H  ins.  in 
diameter,  the  cucilicient  of  friction  0-075,  what  fraction, 


18  moving  OL 
an  lioiir  when 
ue  will  it  move 

friction  'x-ing 
13374.8  ft. 

a  biidy  woigli- 

^tli  of  a  rough 

and  the  base 

,  300ft.-lbs. 

?nd  an  incline 
city  fg  ft.  per 
h  what  space, 
st,  the  friction 
1120f'o2 

[^4c  +pj' 

:o  be  attached 
)f  the  incline, 
1  to  the  other 
.vill   the   traia 


7\ 


ML 

1120 


20  revolutions 
tons  and   the 

nergy  in  fuot- 
H.s'.  30;054. 

ed  a  weight  of 
giilar  velocity 

be   n    ins.    in 
what  fraction, 


EXAMI'IjKS. 


A-ib 


approximately,  of  the  3.")  horse-power  is  expended  in  turn- 
ing the  lly- wheel  ?  Ans.    ^. 

b-i.  In  pile  driving,  38  men  raised  a  ram  12  times  in  an 
hour;  the  weight  of  the  ram  was  12  cwt.,  and  tlie  height 
through  wiiicli  it  was  raised  140  ft.;  tind  the  work  done  by 
one  mall  m  a  minute.  Aiis.  ':)00  ft.-lbs. 

53.  A  liattering-ram.  weighing  2000  lbs.,  strikes  the 
iiead  ol'  a  pile  with  a  velocity  of  20  ft.  per  second;  how  far 
will  it  drive  liie  pile  if  the  constant  resistance  is  10000  lbs.? 

Ans.  1.25  ft. 

54.  A  nail  2  ins.  long  was  driven  into  a  block  by  sue 
cessive  blows  from  a  monkey  weighing  r).01  lbs.;  =ifter  one 
blow  it  was  found  that  the  head  of  the  nail  projected  0-8 
of  an  inch  ai)Ove  the  surface  of  the  block  ;  the  monkey  was 
then  raised  to  a  height  of  l..")  ft.,  and  allowed  lo  fall  upon 
the  head  of  the  nail:  after  this  blow  the  head  of  the  nail 
was  0.40  of  an  inch  above  the  surface;  tind  the  force  which 
the  monkev  exerted  upon  the  head  of  the  nail  at  this  blow. 

Ans.  265.24  lbs. 

55.  The  monkey  of  a  pile-driver,  weighing  500  ll>s.  is 
raised  to  a  height  of  20  ft.,  and  then  allowed  to  tall  upon 
the  head  of  a  pile,  which  is  driven  into  the  ground  1  indi 
by  the  blow;  find  the  force  wliicli  the  monkey  exerted 
upon  the  head  of  the  pile.  Anx.   120000  lbs. 

50.   A  stciini  hammer,  weighing  500  lbs.,  falls  through  a 

height  of  4  ft.  under  the  action  of  its  own.  weight  and  a 

steam    pressure   of   1000   lbs.;    find    the  amount  of   work 

which  it  can  do  at  the  end  of  the  fall. 

/Ih.x.  0000  ft.-lbs. 

57.  The  mean  section  of  a  stream  is  8  scjuare  ft.;  its 
mean  veloeitv  is  40  ft.  per  minute;  it  has  a  fall  of  17J  ft.; 
it  is  required'  to  raise  water  lo  a  height  of  300  ft.  by  means 
of  a  water-wheel  whose  modulus  is  0.7;  how  many  culiic  ft. 
will  it  raise  per  minute  'i  Ans.   13.07  cub.  ft. 


* 


426 


KXA.HPLES. 


58.  To  what  liciglit  would  the  wheel  in  the  last  example 
raise  'i^  cub.  ft.  of  water  pt-r  minute  ?         Ans.   174.if  ft. 

50.  The  meau  section  of  a  stream  is  1|  ft.  by  11  ft.;  its 
mean  velocity  is  --i^  miles  an  hour  ;  tliere  is  on  it  a  fall  of 
fi  ft.  on  which  is  erected  a  wheel  whoso  modulus  isU.T;  this 
wheel  is  employed  to  raise  tiie  hummers  of  a  forge,  each  t.t 
which  weighs  2  Urns,  and  has  a  lift  of  U  ft.;  how  many 
lifts  of  a  hammer  will  the  wheel  yield  i)er  minute!' 

Ans.   H2  nearly. 

()().  In  the  last  example  determine  the  mean  depth  of 
the  stream  if  the  wheel  yiehls  135  lifts  per  nunute. 

Ans.   1.43  ft. 

61.  In  Ex.  59,  how  many  cubic  ft.  of  water  must  descend 
the  fall  per  minute  to  yield  97  lifts  of  the  hammer  per 
''»''>"tt'?  A/is.   ^4.:>3  cub.  ft. 

()--i.  A  stream  is  a  ft.  broad  and  b  ft.  dcej),  and  flows  at 
the  rate  of  c  ft.  per  hour;  there  is  a  fall  of  /i  ft.-;  the  water 
I  urns  a  nuichine  of  which  the  modulus  is  e  ;  find  the  num- 
l)er  of  bushels  of  corn  which  the  machine  can  grind  in  an 
hour,    supposing   that   it   re(iuires   m    units   of   work    i)er 


minute  for  one  hour  to  grind  a  bushel. 


An 


lOOOabchc 


s.  


I'i  X  iiOm 


ly.i.  Down  a  U-ft.  fall  2(K)  cul).  ft.  of  water  descend  every 
minute,  and  turn  a  wheel  whose  modulus  is  O.fi.  The 
wheel  lifts  water  from  the  l)ottom  of  the  fall  to  a  height  of 
54  ft.:  (1)  hov,  many  cubic  ft.  will  be  thus  raised  per 
minute?  (•*)  If  the  water  were  raised  from  the  toj)  of  the 
fall  to  the  same  point,  what  would  the  number  of  cubic  ft. 
1'"^^"  ^^''-  ^l"-*-    (1)  31.1  cub.  ft.;  (•>)  ;{4.:  cub.  ft. 

In  tlie  second  Pase  tlu>  numlHT  of  ciil>.  ft.  of  water  tiikeii  from  tlio 
top  of  tlie  full  l)eint!:  j;  tlie  nunilMT  of  ft.  tliut  will  turn  tlie  wlieel  will 
be  200  -  <: 

U4.  Find  how  many  units  of  work  are  stored  up  in  a 


last  example 
s.   174--if  ft. 

by  11  ft.;  its 

II  il  11  fall  of 

us  isU.;;  tins 

'orge,  each  of 

. ;  how  many 

ite? 

14:i  nearly. 

[?an  depth  of 

lute. 

IS.   1.43  ft. 

must  descend 
hammer  per 
o3  cub.  ft. 

and  flows  at 
t/;  the  water 
ul  the  num- 
grind  in  an 
)f  work  j)er 
lOOOabrhc 

it;  X  aoiii' 

escend  every 
«  O.f).  The 
J  a  height  of 
s  raised  per 
e  top  of  the 
of  cii!)ic  ft. 
.7  cul).  ft. 

iki'ii  friiin  tho 
lilt-  wlii'fl  will 

•ed  lip  in  a 


Ji.XAMI'LhS, 


437 


mill-pond  which  is  lOd  ft.  long,  oO  ft.  i)road,  and  .'!  ft.  deep, 
and  has  a  fall  of  8  ft.  A7is.  7500000. 

05.  There  are  three  distinct  levels  to  be  pumped  in  a 
mine,  tlie  tirst  100  fathoms  deep,  the  second  120,  the  third 
150  ;  ;K)  cub.  ft.  of  water  are  to  come  fnmi  the  first,  40  from 
the  second,  and  (iO  from  the  third  jK'r  minute  ;  the  duty  of 
the  engine  is  TOOOOOOO  for  a  busiiel  of  coal.  Determine  (1) 
its  working  horse-power  and  (2)  the  consumption  of  coal 
per  hour.  Ans.   (1)  191  II.-l'. ;  (2)  5.4  bushels. 

CO.  In  the  last  example  suppose  there  is  another  level  of 
IGO  fathoms  to  be  i)umi)ed,  that  tiie  engine  does  as  much 
work  as  l)eforo  for  the  other  levels,  and  that  the  utmost 
))ower  of  tho  engine  is  275  II.-P. ;  tind  tho  greatest  number 
of  cub.  ft.  of  water  that  can  ijo  raised  from  the  fourth  level. 

A)is.  iO^  cub.  ft. 

07.  A  variiilde  force  has  acted  through  8  ft.;  the  value 
of  the  force  taken  at  nine  successive  equidistant  })oints, 
including  the  first  and  tho  last,  is  in  lbs.  10.204,  !t.804, 
0.434,  U.OOO,  8.771,  8.475,  8.197,  7.937,  7.G92 ;  find  the 
whole  work  done.  Aiis.   70.041  foot-jjounds. 

G8.  The  value  of  a  variabl'»  force,  taken  at  nine  succes- 
sive equidistant  points,  including  the  first  and  the  last 
points,  is  in  lbs.  2.4840,  2.5049,  2.(3391,  2.7081,  2.7726, 
2.8332,  2.8904,  2.9444,  2.9957,  the  common  distance  between 
the  points  is  1  ft.;  find  tho  whole  work  done. 

Alls.  22.0957  foot-pounds. 

09.  A  train  whose  weight  is  100  tons  (including  the 
engine)  is  drawn  by  an  engine  of  150  horse-power,  the  fric- 
tion be''ig  14  lbs.  per  ton.  and  all  other  resistances  neglect- 
ed ;  find  the  maximum  spetd  which  the  engine  is  capable 
of  sustaining  on  a  level  rail.      Jus.  40|;*jf  miles  per  hour. 

70.  If  the  train  described  in  the  last  example  be  moving 
ut  a  particular  instant  with  a  velocity  of  15  miles  per  hour, 


ita 


428 


EXAMPLES. 


iind  tlu'  oiigiiio  working  at  full  power,  wluit  is  tlio  accelera- 
tion at  that  instant  ?     (Call  //  =  •.V>.)  A^s.  AV 

71.  Find  the  horse-power  of  an  engine  i-equired  to  drag  a 
*rain  of  100  tons  up  an  incline  of  1  in  oO  with  a  velocity  of 
30  uiilos  an  hour,  the  friction  being  1400  lbs. 

Alls.  The  engine  must  be  of  not  less  than  4?0f  horse- 
power. This  is  somewhat  ahove  tlie  j)ower  of  most  locomo- 
tive engines. 

72.  A  train,  of  200  tons  weight,  is  ascending  an  incline 
ot  1  in  100  at  the  rate  of  30  miles  per  hour,  tlie  friction 
being  8  lbs.  per  ton.  The  steam  l)eing  shut,  off  and  the 
break  applied,  the  train  is  stopped  in  a  (piarter  of  a  mile. 
Find  the  weight  of  the  break-van,  the  coeflicient  oi'  fric- 
tion of  iron  on  iron  being  f  Ans.  ll^j  tor:  5. 


i.s  tlio  accelera- 
Ans.  T*iV 

uired  to  drag  a 
h  a  velocity  of 

lan  4?0|  horso- 
f  most  locomo- 


iiig  an  incline 
ur,  the  friction 
ut,  off  and  the 
rtcr  of  a  mile. 
Iticient  oi'  fric- 
!.  ll-j*f  tor:?. 


C  H  A  I^  T  E  R    VI. 

MOMENT    OF     INERTIA.* 

224.  Moments  of  Inertia. — 'I'lic  ([uantity  ^mr^  in 
which  vi  is  the  ma.s.s  of  an  element  of  a  IhkIj,  and  /•  it.s 
distance  from  an  axis,  oecnr.s  fre(|ucntly  in  problems  of 
rotation,  so  that  it  becomes  necessary  to  consider  it  in 
detail  ;  it  is  called  /he  niomviit  of  inertia  of  the  body  about 
the  axis  (Art.  218).  Hence,  "moment  of  inertia"  may  be 
defined  as  follows:  If  //le  inasft  of  crn-j/  jiurHdc  (fa  bodij  be 
muUipliod  by  the  squun'  of  itx  dis/aiicrfrom  a  straight  lim, 
tlie  ftum  of  the  products:  so  formed  is  called  the  Moment  of 
InertiAi  of  the  body  about  that  tive. 

If  the  mass  of  every  particle  of  a  body  be  multiplied  i)y 
the  square  of  its  distance  I'rom  a  given  plane  or  from  a 
given  point,  the  sum  of  the  pioducts  .so  formed  i.s  called  the 
moment  of  inertia  of  the  body  with  reference  to  that  plane 
or  that  point. 

If  the  body  be  referred  to  the  axes  olV  and  y,  and  if  the 
mass  of  each  particle  be  multiplied  by  its  two  co-ordinates, 
■X,  y,  the  sum  of  the  j)roduct8  so  formed  is  called  the 
product  of  inertia  of  the  l)ody  about  those  two  axes. 

If  dm  denote  the  mass  of  an  element,  p  its  distance  from 
the  axis,  and  /  the  moment  of  inertia,  we  have 

I=^lPdm.  (1) 

If  the  body  be  referred  to  rectangular  axes,  and  x,  y,  z, 
be  the  co-ordinates  of  any  element,  tiieu,  according  to  the 
definitions,  the  moments  of  inertia  aluiut  the  axes  of  .r,  //, 
!.,  respectively,  will  be 

•  This  term  wbh  Inlrodiiccil  by  KuU.t,  and  lias  now  k"!  '»•"  Ki-nural  ui^o  wlicii 
ev'ir  I):U;'(1  UyuBulM  U  sludioil. 


^ 


430  EXAMPLES. 

2:  (y2  +  «2)  dm,    ^  (z2  4-  .1-2)  dm,    ^  {a^  +  if)  dm.     (2) 

The  moments  of  inertia  witii  respect  to  the  planes  yz,  zx, 
xy  respectively,  are, 

1  x^dm,     X  if  dm,     il  zhlm.  (;{) 

The  products  of  inertia  witli  respect  to  the  axes  y  and  2,  • 
%  and  ./■,  X  and  //,  are 

1  yzdm,     i  zxdm,     }L  xydm,  (4) 

The  moment  of  inertia  with  respect  to  tiie  origin  is 

V  (.,.-'  -I-  //3  +  2^)  ,li,i  =  V  ,a^{„^^  ^5^ 

where  r  is  the  distjinee  of  the  particle  from  tlic  origin. 

Tile  moment  of  inertia  of  a  lamiiiii,  when  the  axis  lies  m 
it,  is  called  a  rcr/inH/ii/ar  iiHiiiifid  nf  inertia,  and  wiien  it  is 
jK'rpendiciilar  to  the  lamina  it  is  called  a  imlur  moment  of 
inertia,  and  the  corresponding  axis  is  called  the  redanyular 
or  the  jiolar  axis. 

The  process  of  linding  nionirnts  and  prodacts  of  inertia 
is  merely  that  of  integration  ;  but  after  this  has  been  accom- 
plished for  the  simplest  axes  possible,  they  can  be  found 
without  integration  for  any  other  axes. 


EXAMPLES. 

I.  I''ind  the  moment  of  ii\ertia  of  a  uniform  rod,  of  nniss 
/«.  and  length  /,  aliout  an  axis  through  its  centre  at  right 
angles  to  it. 

\a'{  t  be  the  distaitco  of  any  element  of  the  rod  from  the 
centre,  and  /(  tiie  mass  of  a  unit  of  leiigtii  ;  then  dm  =  jidx, 
wliieh  in  (1)  gives  for  the  moment  of  inertia  "^iix^dx,  or 

'ir 


=    I     fi.r'hlx, 


planes  ijz,  zx, 

(-5) 
axes  y  and  2, 

(4) 


(5) 

'  origin, 
e  axis  lies  ni 
1(1  wiion  it  is 
»•  moment  of 
c  rectangular 

cts  of  iiicrlia 
i  been  acconi- 
an  be  found 


rod,  of  mass 
litre  at  right, 

rod  from  tlio 
111  dm  =  jidjty 
\ix^dx,  or 


EXAMl'LK. 


431 


remonibfring  that  the  symbol  of  siinimatioii,  1,  iiioliidos 
iuUgralion  in  the  cases  wherein  the  body  is  u  continuous 
mass. 

Hence  /  =  ^^fiP  =  ^\mP. 

If  the  axis  l)e  drawn  tiiroiigh  one  end  of  tho  rod  and 
perpendicular  to  its  length  we  shall  have  for  tlie  inomenl 
of  iiiei  tia 

/  =  ImP. 

2.  Find  the  moment  of  inertia  of  a  rectangnlar  lamina* 
aliout  an  axis  through  its  centre,  parallel  to  one  of  its  sides. 

Ix't  d  and  r/ denote  the  breadth  and  depth  respectively  of 
the  rectangle,  the  former  being  iiarailel  to  the  axis.  Im- 
agine the  lamina  composed  of  elementary  strips  of  length  /> 
liarallel  to  the  axis.  Let  the  distance  of  one  of  them  from 
tiie  axis  be  y,  and  its  breadth  dy  ;  tlien,  denoting  the  mass 
of  u  unit  of  area  by  //,  we  have  dm  —  iihdy,  which  in  (1) 
gives 

I  =  idxfdy  =:  ^Kiif)(P  =  J^mcP. 

If  the  axis  be  drawn  through  one  end  of  the  rectangle,  we 
shall  have  for  the  moment  of  inertia 

/  =  \vuP. 

3.  Find  the  inoment  of  inertia  of  a  circulfir  lamina  with 
respect  to  an  axis  through  its  centre  and  perpendicular  to 
its  surface. 

fiCt  the  radius  —  a,  and  fi  the  mass  of  a  unit  of  area  as 
before,  then  we  have 


nut- 


*  III  nil  cjixi'!*  wi'  r-hiill  iissiimr  IIk^  llilikiic- -  of  tliu  liimiiiiB  or  platos  to  b« 
tuflniU'ginial. 


I'AKALLlCh  AXES. 


4.  Y'wA  tlio  moment  of  iiicrtiii  of  a  circular  i)la(c  (1) 
iil^out  u  diameter  us  an  axis,,  and  (:i)  about  a  tangent. 

Ans.   (1)  \ma^',  {'i)  InutK 

5.  Find  the  moment  of  inertia  of  a  square  plate,  (1) 
about  an  axis  through  its  centre  and  perpendicular  to  its 
plane,  ('-i)  about  an  axis  which  joins  the  middle  points  of 
two  opposite  sides,  and  (3)  about  an  axis  jiassing  through 
an  angular  j)oint  of  the  ])late.  and  perpendicular  to  its 
plane.  Let  a  =  the  side  of  the  plate  and  /t  the  mass  of  a 
unit  of  area. 


(I  II 


(5>)   ,»jm«^;    (:J)  InutK 

0.  Find  the  moment  of  inertia  of  an  isosceles  triangular 
plate,  (I)  about  an  axis  through  its  vertex  and  perpen- 
dicular to  its  plane,  and  (2)  aboiit  an  dxis  which  passes 
through  its  vertex  and  l)isects  the  l)a.se. 

Let  21  =  the  base  and  a  =  the  altitude,  then 


/      ,1  (.r^  +  r')  d,/  dx  =  y  {:u,^  +  i»)  ;  (2)  \n,b\ 

225.  Moments  of  Inertia  relative  to  Parallel 
Axes,  or  Planes. —  The  moment  of  inertia  of  a  bodi/  about 
any  axis  ('*■  rtiual  toils  moment  of  inertia  alioid  a  parallel 
axis  through  tlie  centre  of  urarity  of  the  body,  phis  the 
product  of  the  mast  of  the  body  into  the  square  of  the  dis- 
tance between  the  t(xes. 

Let  the  plane  of  tl-.e  i)aper  pass 
through  the  centre  of  gravity  of  tlic 
liody,  and  l)e  perpendicular  to  the  two 
parallel  axes,  meeting  them  in  0  and 
(i.  and  let  P  be  the  projection  of  any  o 
element   on    the    plane   of   tin.'    paper. 


Fig.gi 


EXAilI'LEti. 


433 


rcular  i)la(i'  (1) 
1  tangent. 

jiiaro  plate,  (1) 
pondicular  to  its 
iiiddle  points  of 
passing  through 
I'MtiiciiUir  to  its 
ji  the  mass  of  a 


m 


a- 


scelos  triiinguiar 
L'X  and  })t'rpon- 
icis  which  pusses 

tlieu 


e    to    Parallel 

(  of  a  body  (ibout 
I  akmt  a  pnraUcI 
•e  body,  plun  fhe 
square  of  the  di.s- 


Take  tlie  centre  of  gravity,  G,  as  origin,  the  fixed  axis 
through  it  jjcrpendicular  to  tlie  plane  of  the  paper  as  the 
axis  of  z,  and  the  plane  through  this  and  the  parallel  axis 
for  that  of  i.r;  and  let/,  be  the  moment  of  inertia  about 
the  axis  thnmgh  0,  /  that  for  the  parallel  axis  through  U, 
a  the  distance,  OG,  between  tiic  axes,  and  (.r,  _(/)  any  point, 
J'.     Then  we  shall  have 

I^  =1.  {.r^  +  tf)  dm  ;  /  =^  1  [(•'•  +  ")-  +  if]  dm.. 

Hence         I  —  1^  =  )l(i.^xdm  +  a'^'^dm  =  lihn, 

since  }Lxdm  =  0,  as  the  centre  of  gravity  is  at  the  origin. 

.-.     1  =  1,  +ahn,  (1) 

which  is  called  llu'  fonindn  (f  irdiic/ioii. 

Hence  tlie  moment  of  inertia  of  a  body  relative  to  any 
axis  can  be  found  when  tbat  for  the  parallel  axis  tiirougii 
its  centre  of  gravity  is  known. 

Von.  1. — The  mouu>nls  cf  inertia  of  a  body  are  the  same 
for  all  parallel  axes  situated  at  tlie  same  distance  from  its 
centre  of  gravity.  Also,  of  ail  parallel  axes,  that  which 
passes  through  the  centre  of  gravity  of  a  body  has  the  least 
moment  of  inertia. 

(',„•,  ;i._lL  is  evident  tiiat  the  same  theorem  holds  if  the 
moments  of  inertia  lie  taken  with  respect  ♦o  i»arallel  planes, 
Instead  of  })arallel  axes, 

A  similar  property  also  connects  the  moment  of  inertia 
relative  to  any  point  with  that  relative  to  the  centre  of 
gravity  of  the  body. 

EXAMPLES. 

1,  The  moment  of  inertia  of  a  rectangle*  in  reference 
to  an  axis  through   its  centre  and  iiarallel   to  one  en. I   is 

*  »!'«  Nolo  to  Ex.  8,  Art.  384  ;  sirlclly  hipcoUiui;.  an  area  Iiuh  b  mi)muiil  (if  iiii.'rtiu 
no  luoic  Uinii  It  Imw  wol^lit. 


m 


4:}-4 


RADirs  OF  (;ynATi()\. 


^\md'' ;  find  the  nioniont  of  inertia  in  roferenco  (;>  a  parallel 
axis  through  one  end. 
From  (1)  we  have 


2.  The  moment  of  inertia  of  an  isosceles  triangle  about 
an  .ixis  through  its  vertex  and  perpendicular  to  its  plane 
;s  \,n  (;jrt2  +  ^2),  (Art.  224,  Ex.  G);  tind  its  moment  about 
a  }iarallel  axis  through  the  centre. 

From  (1)  we  have 

.'5.  Find  the  luomeiit  of  inerfiu  "l\i  circle  about  an  axis 
tlirongh  its  circumference  and  perpendicular  to  its  plane 
(See  Ex.  3,  Art.  224).  Ans.  |w«8. 

4.  Find  the  moment  of  inertia  of  a  square  nbout  an  axis 
through  tiie  middle  point  of  one  of  its  sides  and  perpen- 
dicular to  its  plane  (Ex.  5,  Art.  224).  Jns.  ^^mdK 

226.  Radiusof  Gyration.— Let  k  be  such  a  qtiantity 
that  the  moment  of  inertia  =  mk-,  tlicn  wo  shall  have 


^r~dm  =  VI  k\ 


(1) 


The  distance  k  is  called  the  radius  of  gyration  of  the 
body  with  respect  to  the  fixed  axis,  and  it  denotes  tlie 
distance  from  the  axis  to  tiiat  jjoint  into  wliich  if  tlie  wiiole 
lUiiss  were  concentrated  tiie  niomeiit  of  inertia  wouki  no!,  lie 
altered.  Tiie  /wW  into  whicii  the  hoily  might  be  concen- 
trated, witiiout  altering  its  moment  of  inert i;i,  is  called  the 
ccnln'  ofiijiralinn.  When  the  lixed  axis  passes  through  the 
centre  of  gravity,  [\\v  louifh  /•  ;,nd  tiie  point  of  concentra- 
tion are  called  principal  radiun  and  princiiml  centre  of 
(jijralion. 


ICO  t;)  a  parallel 


triangle  about 
ar  to  its  plane 
moment  about 


about  an  axis 
r  to  its  piano 
A71S.  |»ja*. 

nbout  an  axis 
es  and  perpen- 
ins.  -f^ma\ 

ch  a  quantity 
hall  have 

(1) 

fration  (jf  the 
;  dt'iioto.s  tlic 
li  if  the  wliole 
I  would  110!,  1)0 
lit  1)0  fOMcon- 
,  is  oallod  tbc 
s  tlirouffli  tlio 
of  coiK'ontra- 
pdl  centre  of 


KADI  IS    OF  aVUATIOS. 


4:5: 


\a'\  k\  =  the  prinoi|)al  rinlius  of  <ryration  and  /•,  tlio 
distanoo  of  an  olomcnt  from  tlio  axis  through  tho  centro  of 
gravity;  then  from  (1)  wo  have 


mic' 


1  r'hlm 
=  i;  )\^dm  +  md^,  [by  (1)  of  Art.  a^io] 

from  which  it  aj)poars  that  tho  principal  rndinx  of  (jyratiiin 
is  the  Irasf  radiioi  fur  parallel  axes,  which  is  also  evident 
from  Cor.  1,  Art.  •^'■^5. 

Sen. — III  homogonoous  bodies,  since  the  mass  of  any  part 
varies  directly  as  its  volume,  (1)  may  be  written 


^rMV=  r^-2, 


(••5) 


where  d  T  denotes  the  clement  of  volume,  and  V  the  entire 
volume  of  the  body. 

Hence,  in  homogeneous  bodies,  the  value  of  /.■  is  inde- 
pendent of  the  density  of  the  body,  and  depends  only  on  its 
form;  and  in  determining  the  moment  of  inertia,  we  may 
take  the  element  of  volume  or  weiglit  for  the  element  of 
mass,  and  the  total  volume  or  weight  of  the  body  instead 
of  its  nuiss. 

Also  in  tinding  the  moment  of  inertia  of  a  lamiiui,  since 
/•  is  independent  of  the  thickness  of  the  lamina,  we  may 
take  the  element  of  area  instead  of  the  clement  of  ma.s.<, 
iind  the  total  area  of  the  lamina  instead  of  its  mass. 

From  (1)  we  have 


P  = 


m 


Suuilurly, 


'       m 


(5) 


436 


POLAR   MOMKyr  OF  ISEKTIA. 


hence,  the  square  of /he  •iidim  of  ryrafi'/i  wuli  respect  to  f 
any  axis  equals  //  mor„eHl  oj  inertia  with  respect  A.  the  . 
same  aa  is  divided  by  t/ie  mass.  f 


EXAMPLES. 

1.  Find  the  prin(.'ii)al  nulius  of  gyiation  of  a  straight 
line. 

From  Ex.  1,  Art.  224.  we  have 

J\  =  ^mP; 
therefore  from  (5)  we  have  k^^  =  ^P. 

2.  Find  the  jjrincipal  radius  of  gyration  of  a  circle  (1) 
with  respect  to  a  i)oliir  nxin,  and  (2)  with  resi)e('t  to  a 
rectangular  axis.  Ans.   (1)  ^rt^;  (2)  J(/2. 

;j.  Find  the  principal  radius  of  gyration  of  a  rectangle 
with  respect  to  a  reel  angular  axis.  Ans.    ^iP. 

4.  Find  ti)e  principal  radius  of  gyration  (I)  of  a  S(iuare 
•vith  respect  to  a  polar  axis,  and  (2)  of  an  isosceles  triangle 
with  respect  to  a  polar  axis. 

.I//.V.  fi)K;  {-i)  HW  +  i^)- 

227.  Polar  Moment  of  Inertia.— If  any  thin  plate,  or 
lamina,  he  referred  to  two  rectangular  axes  and  x,  y  be  the 
co-ordinates  of  any  clement,  then  (Art.  224)  the  moments 
of  inertia  about  the  axes  of  .r  and  y  respectively,  are  i  if  dm 
and  i  ,rVy»  ;  and  therefore  the  moment  of  inertia  with 
respect  to  the  axis  drawn  perpendicular  to  the  plane  at  the 
inrersection  of  the  axe<  of  ./■  and  //  is 

-  (.'^  -V  y')  dm. 

Hence  the  polar  moment  o/'  inertia  of  any  lamina  is  equal 
to  the  sum  of  the  nnnnrnts  of  inertia  with  respect  to  any  two 
rectangular  axes,  li/iix/  in  the  plane  of  the  lamina. 


'A. 


j'oi,  \/:   iroMi;.\r  of  i.\/:irriA. 


43: 


w.i/i  respect  to 
Ih  resycd  L  Ike 


on  of  a  straight 


Cor. — For  cvcrv   iwo   i((tun<riiliir  axis    in  tlio  plaiio  ;>f 
i,(;e  lamina,  at  any  |K)iiit,  \m'  Iuivl- 

i  xhlm  +  1  if  dm  —  const. 

that  is,  the  sum  of  tlif^  inomeHts  of  inrr/ia  with  respect  to  n 
pair  of  rectangutay  (i.rcs  is  constant.  Ilenco,  if  one  br  •. 
maximum,  the  other  is  a  minimum,  and  cice  versa. 


n  of  a  circle  (1) 
ilk  rosi)e('t  to  a 
)K:  (•-')K- 

n  of  a  rectangle 
.     J  ns.    ^ip. 

(1)  of  a  S(|uarc 
sosceles  triangle 

)  i  (K  +  ^)- 

ny  thin  plate,  or 
and  X,  y  be  the 
4)  the  moments 
vely,  are  il  if  (tin 
of  inertia  with 
the  plane  at  the 


taminn  is  eqvat 
sped  to  an II  two 
tmina. 


EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  rectangle  ;  .M;  r--  ..(.nt 
to  an  axis  through  its    entrc  and  perpendiculai     >  its  ^  ■ni\< . 

From  Fx.  2,  Art.  224,  the  reetangulur  -ne  t,s  of 
inertia  are 

-,\hu/^  and  ^hmb^ ; 

therefore  tlic  polar  moment   of   inertia  =:  i^^in  {(f''  +  t/^)  ; 

2.  Find  the  moment  of  inertia  of  an  isosceles  triangle 
with  resi)ect  to  an  axis  through  its  centre  parallel  to  its 
iKise,  a  being  the  altitude  and  and  2b  the  base. 

Atis.  ^nia^;  ki^—  ^^aK 

228.  Moment  of  Inertia  of  a  aolid  of  Revolution, 
with  respect  to  its  Geometric  Axis.— Let  the  axis  l)e 
that  of  .*•;  and  let  the  e(|uation  of  the  generating  curve 
be  y  7=  fix).  Let  the  solid  lie  divided  into  an  infinite 
nund)er  of  circular  plates  perpendicular  t((  liie  axis  of 
revolution;  let  the  density  be  unil'orm  and  fi  the  mass  (»f  a 
unit  of  volume  ;  and  denote  l)y  .;•  the  distance  of  the  centre 
of  any  circular  plate  from  tiie  origin.  //  its  radius,  and  dx 
its  thickness  :  then  the  moment  of  inertia  of  this  circular 
plate  iibout  an  axis  through  its  eentiv  and  i)erpendieu]ar  to 
its  plane,  by  (Ex.  3,  Art.  tl\),  is 


mm 


438 


EXAM  I' Las. 


(licrcforo  the  moment  of  inertia  ol'  tiic  whole  soHcl  is 

.       YJ\f{^)Ydx,  (1) 

tile  integration  being  taken  hetween  proper  limits. 

EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  rigjit  cirenlar  cone 
about  its  axis. 

Let  h  —.  the  height  and  h  =  the  radius  of  tlie  bsise ; 
then  tlie  equation  of  the  generating  curve  i^^  U  =  ,  ^'> 
which  in  (J)  gives  for  the  moment  of  inertia, 

^  -  2h*  Jo  ~  ^0' 

—  j^iiiff',  (since  m  =     nhhA- 
Therefore    Z-,2  =  ,yA 

2.  Find  the  moment  of  inertia  (1)  of  a  solid  cyrnder 
about  its  axis,  b  being  its  radius  and  //  its  height,  and  [i) 
of  a  liollow  cylinder,  h  and  b'  being  the  external  and 
internal  radii.  A)is.   (1)  ^mU^;  {'i)  ^m  (I^  +  b'^). 

.'5.  J-ind  the  moment  of  inertia  of  a  paralwloid  about  its 
axis,  h  being  its  altitude  and  b  the  radius  of  the  base. 

229.  Moment  of  Inertia  of  a  Solid  of  Revolution, 
with  respect  to  an  Axis  Perpendicular  to  its  Geo- 
metric Axis. — Take  the  origin  at  the  intersection  of  the 


tmm 


^m 


EXAMl'LKS. 


439 


solid  is 


iniits. 


(1) 


it  circuliir  cone 

i   of   the  biisc ; 

b 
vei.  y  =  ^^,, 


nn 


solid  cyl'nder 
u'iglit,  and  (2) 
'  oxtcrnal  and 
r//(  <i'  +  b'^). 

oloid  ahouf  its 
the  base. 
,        ■rruhb^ 

I  Revolution, 
r  to  its  Geo- 

rsoctioii  of  tiio 


axis  of  rovolntioii  with  the  axis  ahont  which  the  moment 
of  inertia  is  m|iiired  ;  and  denoting  by  x  the  distance  of 
I  he  centre  cd'  any  circular  phite  from  the  ori«:in,  //  its 
ladius  and  dx  its  thickness,  we  liave  for  th"  moment  of 
inertia  of  this  circnlar  plate,  about  a  diameter,  bv  Ex.  4, 
Art.  ^•.>4, 

4    "•'' ' 

therefore  (Art.  -iil'y)  the  moment  of  inertia  of  this  plate 
about  the  i)ara]lel  axis  at  tlie  distance  x  from  it  is 

-  -^  dx  +  TT/t  yix^  dx ; 

therefore  tlie  moment  of  inertia  of  the  whole  solid  is 

'^/'/(f +  ?H''-^'  (1) 

the  integi-ation  being  taken  between  proper  limits. 

EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  right  circular  cone 
about  an  axis  through  its  vertex  and  perpendicular  to  its 
own  axis. 

Let  //  =  the  height  and  b  ■=.  the  radius  of  the  base,  then 
the  moment  of  inertia  from  (1) 

2.  Find  the  moment  of  inertia  of  a  cone,  whose  altitude 
=  //,  and  the  radius  of  whose  base  :=  /;,  about  an  axis 
through  its  centre  of  gravity  and  i)erpendicular  to  its  own 
iixis.  Alls,  ^m  (Ji-i  +  W). 


440 


KXAMl'LKS. 


■*>.  Find  the  monifiit  of  iiit-niii  of  a  paraboloid  of  rcvoln- 
tioii  about  an  axis  tliroiij^b  its  vortex  ami  |H'rpt'iidicMilar  to 
its  own  axis,  tiio  allitiido  boin^'  //  and  tho  I'adiiis  of  tin' 
base  b.  ,        -ftlib'^ 


A  us 


Vi 


(//^  +  :5//2). 


230.  Moment  of  Inertia  of  Various  Solid  Bodies. 


EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  rcctanguhir  parallel- 
.ipiped  about  an  axis  tiiroujrb  its  rcntiv  of  gravity  and  \)&y- 
aliel  to  an  edge. 

Ijet  tiie  edges  l)e  a,  b,  <■ ;  since  a  parallelopiped  may  be 
conceived  as  consisting  of  an  infinite  number  of  rectangular 
hiniiiiiP,  caeli  of  which  lias  the  .same  ratlins  of  gyration 
relative  to  an  axis  perj)endicular  to  its  plane,  it  follows 
that  the  radius  of  gyrarion  of  the  ])arallelopiped  is  the 
same  as  that  of  the  laniinte.  Hence,  the  moments  of 
inertia  relative  to  three  axes  through  the  centre  and  par- 
allel to  the  edges  a,  b,  r,  respect  i  >ely,  are  by  Ex.  1,  Art. 
227,  j\tn  (W  +  (?■),  iV"  («'  +  '■')•  i^'j'"  («^  +  ^'^)- 

2.  Find  the  moment  of  inertia  of  a  rectangular  parallel- 
opiped al>out  an  edge. 

Thi,.  may  be  obtained  immediately  from  the  last  exam- 
l)le  !)y  ubing  Art.  225.  or  otherwise  independently  as 
follows  : 

Take  the  three  edges  rt,  h,  c  for  the  axes  of  x,  y,  z, 
rcsi)cctively  ;  let  }i  be  the  mass  of  a  unit  of  volume,  then 
the  moment  of  inertia  relative  to  the  edge  a  is 

I     I  l'{  f  +  z^)  lU  dy  dz 

vi    '0    ''O 

o 


M(rrW.\     Oh'   I. \  Kill' I A     ')/■•   .1     /,.l.)//.V.l. 


441 


oloiil  of  rovoln- 
iM-pt'iidicMilar  to 
I'  ^il(ii^l^•  of  tlif 

Solid  Bodies. 


ngiilar  parullol- 
[ravity  and  jiar- 

lopiped  may  l)o 
ir  of  rectaiifjular 
ins  of  fifyration 
)laiu',  it  follows 
k'lopiped  is  the 
U'  nionients  of 
:;enti'c  and  i)ar- 
by  Ex.  1.   Art. 

ngular  panilk'l- 

1  the  last  exam- 
idopendently   as 

axes  of  X,  y,  z, 
jf  volume,  then 
t  is 

dz 


')', 


and  iriiniiliirls    I'nr  Ihc   nioincnts  of  inertia  aliiiu!  t lie  edges 
f>  and  r. 

'i'he  Mionicnl  <<\'  inertiji  of  ,i  cuhe  w  iiosc  edge  is  </  with 
respect  to  oni>  of  its  edges  i~  I'nr'  =  piK/'. 

'A.  l-'iiiil  tiic  niiimciit  of  inerti;i  of  a  segment  of  a  spheri' 
relative  to  a  (!iainetcr  parallel  to  the  plane  of  section,  the 
radius  of  the  sphere  oeing  a  and  tiie  distance  of  the  ])Iaia' 
section  from  the  centre  I). 

Alls.   ^„-u  (Ku^s  +  \')ii^f)  +  \()tiV>^—  91^). 

231.  Moment  of  Inertia  of  a  Lamina  nth  respect 
to  any  Axis. —  When  the  moment  of  inertia  of  a  plane 
figure  about  any  axis  i>  known,  we  easily  find  the  moment 
of  inertia  about  iiny  parallel  axis  (Art.  ■^'2."));  also,  when 
the  moments  of  inertia  about  two  rectangular  axes  in  the 
plane  of  the  ligure  are  known,  the  moment  of  inertia  about 
the  straight  line  at  right  angles  to  the  j)lane  of  these  axes 
at  their  intersection  is  known  immediately.  (Art.  227)  ;  we 
now  proceed  to  lind  the  moment  of  inertia  about  any 
straight  line  in  the  jilaiie  inclined  to  these  axes  at  inii/ 
angle. 

Through  any  point,  0,  as 
origin,  draw  two  rectangular 
axes,  OX,  OY.  in  the  ])lane  of 
the  lamina;  ami  draw  any 
straight  line,  O.r,  in  the  ])lane. 
It  is  reiiuired  to  lind  tlie  mo- 
ment of  inertia  about  O.r  in 
U-'-'ns  of  the  moments  of  inertia  about  OX  and  OY. 

|,<'t  P  be  any  point  of  the  lamina,  .r,  //.  its  rectangular, 
aiul  .  6,  its  polar  co-ordinates,  p  =  PM,  and  «  the  angle 
^•OX.  Then  if  /  be  the  monu'iit  of  inertia  of  the  lamina 
relative  to  Ox,  a  and  h  the  moments  of  inertia  relative  to 
the  axes  of  x  and  //  respectively,  and  h  the  product  of 
inertia  relative  to  the  same  axes,  we  have 


Fig.92 


^mm 


44'-e 


I'UlXcll'Ah    AXhS    OF  A    HOhV. 


I  =  i;/>Vm  =  ^  /••-  m\'  (0  —  u)  dm 
=z  i;  {if  cos  (I  —  .'  sill  (()' (hii 

=:  ('(is^  ,t  i  ij'-diii    I    sin-  »c  1  y-ihii  —  'I  sin  <«  cos  «  i  .ri/dni 
=  II  cos-  <c  -f-  A  sill-  ic  —  '^/i  sill  ic  ('lis  »£.  (1) 

If  we  ciioosr  the  ii.M's  sii  (liiit  the  (crm  //  or  }1  xi/dm  =  0. 
t lie  expression  for  y  liecomes  iiiiich  sinijilcr.  Tlio  pair  of 
iixes  so  t'liosen  jire  ciilled  tlie  j/riiicipal  i!.ri's  at  the  point; 
iiiul  the  ooiTeS})on(ling  nionieiits  of  inertia  are  called  I  he 
principal  mom enl.s  of  in ni. in  of  the  lamina,  relative  to  the 
point. 

If  A  and  IJ  represent  these  jirincipal  nioinents  of  inertia, 
(1)  becomcH 

/  =  A  cos^  ,i  +  />'  siii2  «.  (i) 

Hence,  /he  moment  of  inerlia  if  a  la  in  inn  with  rc^tpecf  to 
any  iixis  iliroiKjh.  a  point  maij  tw  found  when  the  principal 
moments  with  respect  to  t/ie  /loiiit  ore  detennined. 

232.  Principal  Axes  of  a  Body. — At  any  point  of  a 
rigid  twdy  ami  in  any  plane  there  is  a  pair  of  principa\ 
axes. 

Let  OX,  OY  (Fig.  03),  be  any  rectangular  axes  in  the 
])lune;  let  0.r,  Oy,  be  anotlier  set  of  rectangular  axes  in 
tlu' same  phme,  inclined  lo  the  tbriner  at  an  angles:  let 
a,  I),  and  //,  as  before,  denote  the  nionients  and  jiroduet  of 
inertia  about  OX,  OY.  and  let  (x',  y')  be  any  ]ioint,  /^ 
referred  to  the  axes  0,r,  Oy.  Then,  using  the  nutation  of 
the  last  article,  we  have 

x'  =  r  cos  {0  —  rt) ;     y'  =  r  sin  (6  —  a)  • 

S  x'y'dm  ~  \)l  r^  sin  "2  {0  —  «)  dm 

=  cos  2«  1  r^  sin  0  cos  0  dm 

-  I  sin  3fc  X  ;•»  (cos2  0  -  sin»  6)  dm. 

Putting  this  =z  0,  and  sohiiig  for  tc.  we  obtain 


a  cos  ^c  !•  .ri/dm 


V  i;  xi/(hn  =0. 
,     Tlic  pair  of 

ill,  tlie  point; 

are  called  I  he 
relative  to  the 

ents  ot  inertia, 

ivi/h  respect  to 
>i  the  principal 
ited. 

any  point  of  a 
ir  of  principa[ 

iv  axes  in  the 
iigiilar  axes  in 
in  aiifi'le  «  :  let 
iiul  ])r()(liiel  of 
any  ]ioiirl,  /^ 
he  ii'-tatiou  of 


-  «) ; 

Im 
0  dm 

—  sill*  6)  dm. 

tin 


(1)        1 


Tiiia:h:  i'iii\<ii'Ai.  axks. 


44; 


tan  2a 


22  /■'  sill  ft  cos  H  dm 
^^?lcos'  H  -~^\n'  H)dm 

2^.ri/dm       _      2h 


(1) 


As  the  tanpent  of  an  •mgle  may  have  any  value,  positive 
or  negative,  from  G  to  oo ,  it  follovs  that  (1)  will  a) way, 
.■rife  a  real  value  lor  itt,  so  there  is  always  v,  set  of  princi- 
pal axes  ;  that  is,  ,'/  ererj/  p')uit  in  a  bodi/  i/iere  exi.<tii  one 
pair  of  recta>i(/i/tar  axes  fir  ir/iirh  t/n'  i/hnntiii/  I,  or 
^  xi/  dm  =  0. 

Cou. — It  may  also  lie  sliou!,  that  at  every  ])oint  o.  a 
rigid  hody  tliere  are  three  axes  at  right  angles  f-i  one 
another,  for  wiiieh  ijie  ju-oduets  of  inertia  vanish.* 


♦  Let  a,  6,  c.  bo  llie  momoiit»  of  liiorliii  iiboul  three 
axes,  ox,  OY,  OZ,  at  right  angles  to  one  aiiotlier ;  d.  e, 
/,  till'  iiroduetn  of  inertia  (i' >«?/?,  "i-iuzi;  "Hmxi/y  re- 
f|ieclivel,vl.  Let  Ox  be  any  line  drawn  i1ii(jii!,'Ii  the 
origin,  making'  antjIi^H  »,  /?,  j ,  with  the  en  iinliiiate 
axes 

Let,  OL,  LM,  MP,  be  the  co  ordiiiiites  x.  y,  z,  of  any 
point  1'  of  the  body  at  which  an  elcnieui  of  mass  m  is 
Hiiiialeci.    nraw  I'N  i^erperidienlar  to  Or, 

ProjecliriLr  the  broken  line,  OLMP,  on  ON,  (Art. 
lOS).  we  have 

.!■  COS  II  I  y  cos  0  +  z  cos  j  ; 


Fig.ora 


ON 

iilfo  OI'-       .I-''  +  I/'  +  z\     niid     1     -  cos'  II  +  cos'  /?  +  cos*  y. 

The  moment  of  inertia  I  abont  Or       I'mPN' 
"  I'm  (OP"  -  ON') 

=  i'm  [,!■'  f  ;/•  1   ■'  -  (.1'  cos  It  I  y  cos  0  +  z  cos  >)''] 
=  Sto  [(.!'■  + y'N  j'i(cos'  .r4 cos' i9  +  cos''  ))—(.!■  cos  H  +  y  cos /?  +  «  cos  ))>] 
=  i'm(y»  I-  z'\  ma'  n  +  >:tn  (?'  +  x,'')  cos»  0  +  Iw  (,/■'  ♦■  y')  cos"  ; 

—  Hyimyz  cos  0  cos  ;  —  aiiwi.c  cob  j  cos  n  -  Sim  cos  u  cos  // 
=  a  cos"  II  (-  h  cos"  0   ^  r  cos"  ;  -  !J</  iios  //  cos  j 

-  'if.  cos  )  cos  II  -   -y  cos  II  cos  0.  il) 

To  n^present  this  ireometrically,  takt!  n  point  Q  on  ON  ;   and  'ct  Its  distance 
from  O  be  /■,  and  Its  eo ordinnleH  be  :r,,  y,,  z,.    Then 

8!,  =  rcoB  «,    Vi  ""  /"cos/?,    S|  _  r COB  v. 


■M 


% 


444 


THRKK  rnrxriPAr,  axes. 


Sni. — In  Jtiniiy  Ciisos  tlic  position  of  tlio  prinoipal  axes 
can  bo  seen  iit  once.  Supj'ose,  for  example,  \Vf  wish  the 
lu'incipal  axis  for  a  rectangle  when  tiie  jjiven  point  is  tlie 
centre.  Draw  thi-ough  the  centre  straigiit  lines  parallel  to 
the  sides  of  the  rectangle  ;  then  these  will  be  the  principal 


Therefore  (!)  becomes 


ax,'  +  6j/,'  +  «,»  —  'i(ly,z,  -  %ez,<e,  —3f'.i-,t/, 


But  the  uiuation 


2(/y,i,  -  'iez,x,  —  yx,i/,  n  1, 


(2) 


(3) 


(IcnolCB  an  elliiiKoiil  whonc  centre  is  iit  () ;  heciiufe  a.  A,  e  are  necessarily  poBilivc, 
Hince  a  moineiil  of  inertia  ii*  csfenliiillv  positive,  being  llie  sum  of  a  ninnlxr  (jf 
i^qnareH,    If  then  (^  in  a  point  on  tliis  ellippoid,  {'i)  1)econieH 


I       >:ml'>i' 


1 


or  the  moment  of  inertia  about  any  line  llirougli  (),  is  nioasurerl  by  the  Kcpiare  of 
the  reciprocal  of  the  radius  vector  of  this  ellipsoid,  which  eoinciiles  with  the 
line. 

This  Is  called  the  momen/al  elHpsold,  and  was  first  used  by  Cauehy  K'j-ircixeK  tie 
Math..  Vol.  II.  It  has  no  i)hysical  existence,  but  is  an  artifice  to  hrinj.;  under  the 
methods  of  (,'eouietry  the  properfii's  of  nionienls  of  iiiorliM.  The  luonieutal  ellip- 
sold  liHs  a  definite  form  for  e\  ery  imint  of  a  rl^'id  body. 

Now  every  clli|>soid  has  three  axes,  lo  which  if  it  is  referred,  ilie  eoeflliieiiLs  (jf 

yz.  !..i\  xy  vanish,  and    tlierefore  (:!).   when  tninsl'oiuio<l  to  these  axes  takes  the 

form 

A.r,»  +  By,'  -  C;,'  =  1; 


and  hence  (I)  or  (i)  when  referred  to  these  axes,  becomes 
I  =  A  cos"  (I  r  B  cos'  /V  1^  C  cos'  ) , 


(I) 


(K) 


where  A,  H,  C,  ore  the  moments  of  inertia  of  the  body  about  these  a\-es. 

When  three  reeianirulrir  axes,  ineetiiifjin  a  u'lven  point,  are  eljosen  so  ihai  ibe 
nroducts  of  inertia  all  vanish.  Iliey  are  lulled  thc^  iivincifml  a.rei-  at  ;he  givei. 
Iiidnl. 

The  three  plain's  ilM'ouj.'h  any  two  principMl  nses  are  (ailed  Ilic  /»-inrt/Hil  /)/nn,N 
al  the  Klven  ])olnt. 

The  moinentB  of  inertia  about  the  prln<ipiil  iixes  ai  any  pointare  called  lhe;jHft- 
cipal  momtnlKo/liierlia  M  that  poini. 

If  the  iliiee  prinri|ial  moments  of  inorlla  of  a  body  are  e(iuiil  lo  one  anotlKT,  the 
ellipsoid  (I)  becomes  a  spiien',  since  A  H  --  (';  and  therefore  the  moment  of 
inertia  about  every  other  axis  Is  equal  to  these,  for  (5)  hccotius 

1  -  A  (cos"  (I  +  COS'  /3  +  COB'  ))  :-  A ; 

and  every  axis   is  a  |)rinilpal  axis.     (See  llonlb's  nip;id  nynaiules,  p   Iv;,  I'rIceN 
Anal.  Mech  «,  Vol.  II,  p.  IWi,  Pirle's  Itlgi.l  Dynamics,  p.  "tl.  ete.i 


THREE  I'KiyciPAI.   AXES. 


44ri 


lirinc'ipal  axofl 
L',  wo  wish  till! 
in  point  is  tlic 
inos  purailul  to 
e  the  priiK'ipal 


(2) 


=  1,  (3) 

necessarily  poeltivc, 
juni  <il  a  miiiibi'r  of 


red  by  tlie  square  of 
1  ci)iiiciilo!>   with   tlie 

Cancliy.  Exircixm  ile 
(■('  ti>  l:riii^;  under  tlic 
Till'  iiiii'iioiital  ('llip- 

I'll,  llll'  I'licfllricllts  of 
IliCHC  uxi'M  takes  the 

(1) 


these  axes. 

re  ehosoii  so  ihnt  the 

)(tt  (Vrn   lit    ;he   i^lven 


axes;  because  for  every  element,  thii,  on  one  side  of  tiie 
axis  of  X  at  the  })oint  {.r,  //),  tliere  is  another  element  of 
ecjual  mass  on  Die  otlier  side  at  tiie  point  {x,  — y).  lleiiee, 
1  xy  dm  consists  of  terms  wiiicli  may  lie  arranged  in  pairs, 
so  that  the  two  terms  in  a  pair  are  niimericully  eipial  but 
of  opposite  signs  ;  and  llierefore  1 .///  (/u)  =  0. 

Again,  if  in  any  uniform  body  a  straiglit  line  can  be 
drawn  with  respect  to  which  the  body  is  exactly  symmetri- 
cal, this  must  be  a  principal  axis  at  every  point  in  its 
length.  Any  diameter  of  a  uniform  circle  or  sphere  or  the 
axis  of  a  parabola  or  ellii>se  or  hyperbola  is  a  principal  axis 
at  any  point  in  its  line;  but  the  diiigoiial  of  a  rectangular 
phite  is  not  for  this  reason  a  i)rincipal  axis  at  its  middle 
point,  for  every  straight  line  drawn  i)er])endicular  to  it  is 
not  equally  divided  liy  it. 

Let  the  body  l)c  symmetrical  about  the  plane  ()f  xi/,  then 
for  every  element  (/id,  on  one  side  of  the  i)lane  at  the  point 
(x,  y,  z),  tliere  is  another  element  of  e(|ual  mass  on  the 
other  side  at  the  point  (,r,  y,  —z).  Hence,  for  such  a  body 
i  xz  (lilt  —  0  and  1  t/z  dm  —  0.  If  the  body  be  a  lamina  in 
the  plane  of  xi/,  then  z  of  every  element  is  zero,  and  we 
liave  again  1  .rz  dm  =  0,  2  yz  dm  =  (I. 

Thus,  in  the  case  of  the  ellipsoid,  the  three  })rincipal 
sections  are  all  jdanes  of  symmetry,  and  therefore  the  three 
axes  of  tiie  ellipsoid  are  princifial  axes.  Also,  at  every 
jxiiiit  in  a  lamina  one  principal  axis  is  the  perpendicular  to 
the  plane  of  the  lamina. 


il  I  he  /ttinci/xil  planet 

lint  are  inllei!  lhe;jHn- 

iial  lo  line  aniilher,  the 
refiire  I  lie  iiiiiniiiit  of 


)ynnniies,  |i   I'J,  TilceN 

I'll'.  I 


K  X  A  M  P  U  E  S  . 

1.  Find  the  moment  of  inertia  of  a  rectangular  lamina 
alioul  a  iliagonal. 

l''roni  Ex.  'i,  Art.  5J'M,  the  moments  of  inertia,  about  two 
lines   through    the   centre   jiarallel  to  the  sides  (principir 
luyuieuts  of  inertia)  are 


T 


446 


EXAMPLES. 

^^md^    and     ^^m¥\ 


when'  h  and  d  hit  the  l)roadth  and  deptli  rospectivelv. 

Also,  if  «  he  the  angle  whicii   the  diagonal  niake.s  with 
the  side  b,  we  have 

'/2  -  b^ 


W  +  iP' 


eos'  «  — 


i^  +  ,/2' 


f^iihstituting  tliese  values  for  A,  B,  sin^ ,«,  cos^  «.  in  (2)  of 
Art.  '^31,  we  have 

2.  Find  the  moment  of  inertia  of  an  isosceles  triangular 
plate  ahont  an  axis  through  its  eentre  and  inclined  at  an 
angle  «  to  its  axis  of  syrametrv,  a  heing  its  altitude  and  2b 
its  base.  An.s.  ^m  (\a^  cos^  «  -f  l/^  sin^  «). 

.'5.  Find  the  moment  of  inertia  of  a  square  plate  ahout  a 
(.iagonal,  a  heing  a  side  of  the  S((uare.  Jus.  ^\iiia\ 

233.  Products  of  Inertia.— The  value  of  the  product 
of  inertia  iit  any  point  may  be  made  to  depend  on  the  value 
of  the  product  of  inertia  for  panillel  axes  through  the  cen- 
tre of  gravity.  Let  {.i;  //)  be  the  jjositioii  of  any  element, 
dm,  referred  to  axes  through  any  as.signed  p.)int  ;  {x',y')  the 
position  of  the  element  referred  to  pirall.l  axes  through 
the  centre  of  gravity,  and  (//,  k)  the  centre  of  gravity 
referred  to  the  Hrst  i)air  of  axes.     Then 

,r    .-    ./    +    h,  ,/    =    _,y'    -I-    /•   ; 

(hevefore       X  ./■■■/ ^/w  =  X  ^,.'    ,    h)  (if  -\-  h)  dm 

z=  V  j;'_//'  ,/„!    I    ////  l,dm,  (1) 

Hlicc  i.  tux'  :-^  0,  and  i-  mi/'  ._;  0. 


i 


ectively. 

1  nuikcs  with 


P 


H.  in   {2)  of 


P 


OS  triiiii(,niliir 
icliiu'il  at,  an 
itudo  and  2b 
■  b'^  sin^  a). 

•lato  about  a 


)IX. 


12 


ma' 


f  tlu'  product 
on  the  value 
ugh  tlie  ccii- 
any  oleniont, 
t;  {x\y')\\w 
ixes  through 
'e  of  gravity 


(0 


EXAMl'hES. 


Ul 


ScH. — By  (1)  we  niuy  often  liiid  tlio  product  of  inertia 
I'oi  an  a>:oi^;:.'u  origin  and  axes.  Tlius,  suppose  we  require 
tlic  product  of  iiurtia  in  tlic  case  of  a  rectangle,  when  the 
origin  is  at  tiie  corner,  and  the  axes  are  tlie  edges  wliich 
meet  at  that  cornei.  Hy  Art.  i'S^,  IScli.  wo  have  l.xy'din 
=  0;  tlierefore  from  (1)  we  have 

l..ti)dni  =z  hk^din  ; 

and  as  h  and  h  are  known,  being  half  the  lengths  of  the 
edges  of  tlie  rectangle  to  which  they  are  respectively 
parallel,  the  product  of  inertia  is  known. 

EXAMPLES. 

Find  the  ex])resBions  for  the  moments  of  inertia  in  the 
following,  the  bodies  being  Kui)posed  homogeneous  in  all 
cases. 

1.   The  moment  of  inertia  of  a  rod    of  length  d,  Avith 

respect  to  an  axis  perpendicular  to  the  rod  and  at  a  distance 

(/  from  its  middle  i)oint.  .  /  a-  \ 

A, IS.  ini^^-^  +  d^y 

'i.  The  moment  of    inertia  of  an  arc  (,f  a  circle  whose 

radius  is  ((  and  whieli  subtends  an  angle  2«  at  the  centre,  (1) 

about  an  axis  through  its  centre  perpendicular  to  its  plane, 

{•I)  al)out  an  axis  through  its  middle  point  perpendicular  to 

its  plane,  (■'})  about  the  diameter  which  bisects  the  arc. 

(        /i\       •)     /o\  o     /i     sin  «\         .  .       /       sin  3«\  ^2 
Ans.   {\))i\a'\  (2)  2//i  (1 Iff.^;  (3)  m  f  1 — ».. 

;).  'file  inomenl  of  inertia  of  the  arc  of  a  comjilete 
cycloid  whose  length  is  n  with  respect  to  its  base. 

4.  The  moment  (  f  inertia  of  an  equilateral  triangle,  of 
side  II.  relative  to  a  line  in  its  plane,  parallel  to  a  side,  at 
the  distance  d  from  its  (ientre  of  gravity. 

Ans.  m  r^'   -I-  iP\. 


448 


EXAMPLES. 


5.  Given  a  triangle  whose  sides  are  a,  b,  r,  and  wliose 
perpendiculars  on  these  sides,  from  the  opposite  vertices 
arejy,  g,  r,  respectively;  find  tlie  moment  ol"  inertia  of  tlie 
triangle  about  a  line  drawn  tlironjrh  eaeli  vertex  and 
parallel  respcc  :vely,  (1)  to  the  side  a,  (2)  to  the  side  b,  (3) 
to  tlie  side  c.  Jus.   (1)  ^m])^;  (i)  ^mif  ;  (3)  Imr'-. 

(').  Find  the  moment  of  inertia  of  the  triangle  in  the  last 
example  relative  to  the  three  lines  draun  through  the 
centre  of  gravity  of  the  triangle  and  parallel  res])ectively 
to  the  sides  a,  b,  c.  Ana.  -^nip^ ;  ^\»irf  ;  -iginr'. 

7.  Find  the  moment  of  inertia  of  the  triangle  in  Ex.  5, 
relative  to  the  three  sides  a,  b,  c,  ivspectively. 

v/n.v.  ^?njP;  }i)i'/~;  |?«r*. 

8.  The  moment  of  inertia  of  a  riglit  angled  triangle,  of 
hypothenuse  c,  relative  to  a  perpendicular  to  its  plane 
passing  through  the  riglit  angle.  J«,v.  ^mc^. 

',K  The  momeni  of  inertia  of  a  ring  whose  outer  and 
inner  radii  aw  a  and  b  respectively,  (1)  with  respect  to  a 
polar  axis  through  its  centre,  and  (2)  with  respect  to  a 
diameter,  Ans.   (1)  |«;  {a^  +  b^)  ;  (2)  im  (a^  +  V). 

10.  The  moment  of  inertia  of  an  ellipse,  (1)  with  respect 
to  its  major  axis,  (2)  with  resjiect  to  its  minor  axis,  and  (3) 
with  respect  to  an  axis  through  its  centre  and  perpendicular 
to  its  plane. 

An,^.    (I)  \..,li^\    {■).)  \mifi;    (3)"  \m  {<t^  +  li^). 

11,  The  moment  of  inertia  of  the  surface  of  a  sphere  of 
radius  a  about  its  diameter.  yl;/.s\  ^nia', 

\'i.  The  moment  of  ineui-inf  a  right  jirism  whose  base 
is  a  right  angled  triangle,  wiili  respect  to  an  axis  passing 
through  the  centres  of  gravity  nl'  the  ends,  the  sides  con- 
laining  the  right  iingle  of  tlic  triangular  base  being  a  and  b 
and  the  height  of  the  jirism  r.  Anti.  -^m  (a*  +i'). 


,  c,  .111(1  whoso 
(posite  vortices 

inertia  of  tin; 
h    vertex    and 

tlie  side  1),  (;}) 
2  :  (3)  \mr\ 

gle  ill  tlie  last 
I  through  the 
lei  res]ieetively 

angle  in  Ex.  5, 


!-  m<i^ 


nir'. 


led  triangle,  of 
to  its  plane 
Ans.  ^mc^. 

[)se  outer  and 
li  respect  to  u 
I   respect  to  a 

)  with  respect 

ir  axis,  and  (3) 

perpendicular 

7)1  {d^  +  //). 

of  a  sphere  of 
Alls,  jfiiia'. 

m\  whose  hase 
in  axis  passing 
the  sides  con- 
being  a  and  d 


EXAMPLES. 


449 


13.  The  moment  of  iuortia  of  u  right  prism  whose  height 

is  c,  about  an  axis  passing  through  the  centres  of  gravity  of 

the  ends,  the  base  of  the  prism  being  an  isosceles  triangle 

whose  base  is  a  and  height  b.  ,        .  id^       d*\ 

Ans.  i  (f  +  3  j  w  . 

14.  The  moment  of  inertia  of  a  sphere  of  radius  «,  (1) 
relative  to  a  diameter,  and  (2)  relative  to  a  tangent. 

Ans.  (1)  \md~\  (2)  fma^. 

15.  The  moment  of  inertia,  about  its  axis  of  rotation,  (1) 
of  a  prolate  spheroid,  and  (2)  of  an  oblate  spheroid. 

Ans.   (1)  \mtl^;  (2)  f/wal 

16.  The  moment  of  inertia  of  a  cylinder,  relative  to  an 
axis  perpendicular  to  its  own  axis  and  inter  •  '>  ,'  it,  (1)  at 
a  distance  c  from  its  end,  (2)  at  tlie  end  of  J:\>  .5,  and  (3) 
at  the  middle  point  of  the  axis,  the  altitude  of  the  cylinder 
being  h  and  radius  of  its  base  a. 

Ans.   (1)  {ma?  +  ^m  {h^  -  She  +  :!r') ; 

(2)  ^m  (3rt2  +  4^2) ;  (3)  ^m  {}fi  +  Sa^). 


17.  The  moment  of  inertia  of  an  ellipse  about  a  central 
radius  vector  /',  making  an  angle  «  with  the  major-axis. 


A  ns.  *m  — r-  • 


18.  The  moment  of  inertia  of  the  area  of  a  parabola  cut 
oflF  by  any  ordinate  at  a  distance  r,  from  the  vertex,  (1) 
about  the  tangent  at  the  vertex,  and  (2)  about  the  axis  of 
the  parabola. 

Ans.  fmx*;  (2)  \my^  where  y  is  the  ordinate  correspond- 
ing to  X. 

19.  The  moment  of  inertia  of  the  area  of  the  lemniscate, 
7^  =  d^  cos  W,  about  a  line  through  the  origin  in  its  plane 
and  perpendicular  to  its  axis,  Ans.  ^m  (3n-.-f-  8)  a*. 


450 


EXAMPLES. 


20.  The  moment  of  inertia  of  the  ellipsoid, 


X8 


;/''       z^ 

+  i-^? 


1. 


about  the  axis  a,  b,  c,  respectively. 

Am.  (1)  im  (bi  +  6-2)  ;  (2)  {m  (c»  +  a»)  j 
(3)  \m  (a^  +  b% 


n  (c'  +  a') ; 


CHAPTER    VII. 

ROTATORY    MOTION. 

234.  Impressed  and  Effective  Forces.— All  forces 
acting  on  a  body  H>er  tlian  the  mutual  actions  of  the 
particles,  are  called  tlie  Jmpresfied  Forces  that  act  on  the 
body. 

Thus,  when  a  ball  is  thrown  in  vacuo,  the  impressed 
force  is  gravity ;  iful)all  is  rotating  about  a  vrlical  axis, 
the  impressed  forces  arc  gravity  and  the  reaction  of  tlie 
axis. 

The  impressed  or  external  forces  are  the  cause  of  the  motion  and  of 
nil  tiie  otlier  forces.  Which  are  the  hiipressed  forces  depends  u])()n 
the  partieulnr  svsleni  wliich  is  under  consideration.  The  same  force 
may  he  external  to  one  system  and  internal  to  another.  Thus,  the 
pressure  between  the  foot  of  a  man  and  the  deck  of  a  ship  on  which 
he  is,  is  external  to  the  ship  and  also  to  the  man  and  is  the  cause  of 
liis  own  forward  motion  and  of  a  slight  backward  motion  of  the  ship ; 
but  if  the  man  and  ship  are  considered  as  parts  of  one  system  the 
pressure  is  internal. 

When  a  particle  is  moving  i?,s  part  of  a  rigid  body,  it  is 
acted  on  by  the  external  impressed  forces  and  also  by  tiie 
molecular  reactions  of  ihe  other  particles.  Now  if  tliis 
])arti('le  were  considered  as  separated  from  the  rest  of  the 
l)ody,  and  all  the  forces  removed,  there  is  some  one  force 
which,  singly,  would  move  it  in  the  same  way  as  before 
This  force  is  called  the  Effective  Force  on  the  particle;  it  is 
evidently  the  resultant  of  the  impressed  and  molecular 
forces  on  the  ])article. 

Thus,  the  effective  force  is  that  oart  of  the  impressed  force  wliicli 
is  effective  in  causinc:  actual  inotioii.  It  is  tlie  force  which  is  required 
for  producing  the  deviation  from  the  straight  line  and  the  chaiige  of 


452 


D  M  LKMH  h'li  T  VV  PI{I.\Cll'LE. 


velocity.  If  a  pnrticlo  is  rt;volving  with  -onstant  vel  icily  rounj  a 
fixed  axis,  the  f'fti'ciivi^  force  is  tlii^  centripc.  .1  force  (Art.  li)H).  if  a 
heavy  limly  falls  without  rotation,  the  whole  force  of  gmvity  is 
effective  ;  but  if  it  is  rotatinji  al)out  a  liorizontal  axis  the  weight  goes 
partly  to  balance  the  pressure  on  the  axis. 

If  we  suppose;  the  iiiii'ticlo  of  muss  m  lo  he  at  the  point 
(.r,  y,  z)  at  any  timo.  /.  ami  resolve  the  forces  aetinjf  on  it 
into  vlie  three  axial  eonij)onents,  X,  )',  Z,  the  motion  may 
be  found  [Art.  1G8  {'i)\  by  solving  the  simultaneous  equa- 
tions 

-i;    '«:7^  =  ^  ;    '"7,72  =  ^-  (1) 


m 


(ir^ 


'"  Tm 


If  we  regard  a  rigid  body  as  one  in  which  the  ])artic]e8 
retain  invariable  jiositioiis  with  respect  to  one  another,  so 
that  no  external  force  can  alter  them  (Art.  -13),  we  might 
write  down  the  equations  of  the  several  particles  in  accord- 
ance Avith  (!),  if  ail  the  "jrces  were  known.  Such,  how- 
ever, is  not  the  case.  We  know  nothing  of  the  mutual 
actions  of  the  i)artieles.  and  consecnu'Utly  cannot  determine 
the  motion  of  tlie  body  by  calculating  tiie  motion  of  its 
jiarticles  separately.  When  there  are  several  rigid  bodies 
Avhich  mutually  act  and  react  on  one  another  tiie  problem 
becomes  still  more  conuilicated. 


235.  D'Alembert's  Principle.*— By  IVAlembert's 
Principle,  however,  all  the  necessary  equations  may  be 
obtained  witliout  writing  down  the  eqtuitions  of  motion  of 
tiie  several  |)articles.  and  without  any  assumption  as  to  tiie 
nature  of  the  iniitual  actions  exceiit  tlie  following,  whicli 
int.y  lie  regarded  as  a  natural  e(mse(iiience  "f  the  laws  of 
motion. 

Till'  iiili'iiud  KctioitK  ami  rracfiims  of  mii/  siisfnn  of  riijld 
Ooi/ii'n  in  motion  a  •"  in  I'l/iiilibriitin  nmonij  llii'iiisi'lris. 


*  IntrcKlucod  by  D'Alunibcrt  in  1T43. 


D 'ALEMliEliT'S   l'Ji'I.\( 'll'LK. 


453 


vericily  rouAjj  a 
[Art.  198).  If  a 
ce  of  gmvity  is 
the  weight  goes 


iit  the  point 
;  iietin<(  on  it 
e  motion  may 
taneous  equa- 

=  Z.  (1) 

I  the  ])article8 
ne  another,  so 
13),  we  might 
2les  in  accord- 
.  Such,  how- 
f  the  mutual 
not  determine 
motion  of  its 
il  rigid  bodies 
T  tlie  problem 


iVAlembert's 
tions  nuiy  1k' 
s  of  motion  of 
»tion  as  to  tlie 
lowing,  wiiit'h 
of  the  laws  of 


ilsfnii  (if  fluid 
I'liisi'lrcs. 


The  axial  accelerations  of  the  })article  of  mass  m,  which 

(I^X      il^U      (t^Z 

is   moving   as   i)art    of  a   rigid    l)ody,   are    -y.^ ,     .."J,     .  .^■ 

Let/ bo  their  resultant,  then  the  effective  force  is  measured 
Ijy  vif.  Let  /•'and  R  be  the  lesultants  of  the  imi)ressed  and 
molecular  forces,  respectively,  on  the  particle.  Then  inf 
is  the  resultant  of  /'and  H.  Hence  if  inf  be  reversed,  the 
tl)ree  forces,  F,  R,  and  //;/,  are  in  e(piilibrium. 

The  same  reasoning  may  be  applied  to  every  ])arti(le  of 
each  body  of  the  system,  thus  furnishing  three  groups  of 
forces,  similar,  respectively,  to  /',  R,  and  nif;  and  these 
three  groups  will  form  a  system  of  forces  in  ecpiilibrium. 
Now  by  D'Alembert's  principle  the  group  R  will  itself 
form  a  system  of  forces  in  ecpiilibrium.  Whence  it  follows 
that  the  group  F  will  be  in  equilibrium  with  the  group  nif. 
Hence, 

//■  forces  equal  and  exactly  opposite  to  tlie  effective  forces 
were  applied  at  each  particle  of  tlie  sjistein,  they  woidd  be 
in  equilibriidii  with  tlie  imjrressed  forces. 

That  is,  D\itembert's  principle  asserts  that  the  whole 
effective  forces  of  a  system  are  together  equivalent  to  the 
impressed  forces. 

ScH. — By  this  principle  the  soluticm  of  a  problem  in 
Kinetics  is  reduced  to  a  problem  in  Statics  as  follows:  We 
lirst  choose  the  co-ordinates  by  means  of  which  the  position 
of  the  system  in  space  may  be  fixed.  We  then  express  the 
eifective  forces  on  each  element  in  terms  of  its  co-ordinates. 
These  effective  forces,  reversed,  will  be  in  ecpiilibrium 
with  the  given  impressed  forces.  Lastly,  the  equations  of 
motion  for  each  1)ody  may  be  formed,  as  is  usually  done  i?i 
Statics,  by  resolving  in  three  directions  and  taking  mo- 
ments aliout  three  straight  lines.  (See  Konth's  l{igid 
Dvnamics.  Pirie's  Rigid  Dynamics,  Pratt's  Mecli's,  Price's 
Anal.  Mech's,  Vol.  IL) 


454 


HOTATJOS   OF  A    liKlIh   UODY. 


236.  Rotation  of  a  Rigid  Body  about  a  Fixed 
Axis  under  tl:;e  Action  of  any  Forces.— Let  tiny 
ItlaiK'  passing  tlinnigli  tlie  axitj  of  rotation  and  lixed  in 
space-  l)e  taken  a.s  a  ])lani  of  rotVreniv.  Let  vt  be  the  mass 
of  any  element  of  the  body.  /'  its  disianee  from  tlie  axis, 
and  0  the  angle  wliieh  a  plane  throngh  the  axis  and  the 
element  makes  with  tiie  plane  of  reference.  * 

Then  the  veloeitv  of  ///   in  a  direi'tion  jierpendicular  to 

(10 


the    plane   containing   the   element   and   the  axis  is  r 


ill 


The   mODioit   of  the   momentum*  of    this   particle    about 

the  axis  is  w/-'-'  ,, .     Hence  tlie  moment  of  the  momeutu  of 
at 


all  the  particles  is 


E  m/-« 


(Id 
lit' 


(1) 


Since   the  particles  of  the   body   are  rigidly  connected, 

•    •  '10  .  .  ..  1   •      , 

it  IS  clear  that  -7,  is  tiie  same  lor  everv  particle,  and  is  the 
(//  .-1 

angular  velocity  of  the  body.      Hence  tlie  moment  of  the 

momenta  of  all  the  part iden  (f  titc  lioihi  abintt  tlie  ajrin  is  tlie 

moment  of  inertia  of  the  l)ody  about  the  axis  multiplied  bij 

the  angular  relociti/. 

The  acceleration  of  m  perpendicular  to  the  direction  in 

iPO 
whicii  /•  is  measured  is  r  r,.,,  and  therefore  the  moment  of 

(tt^ 

.  .        iPn 

the  moving  forces  of  h/  about  the  axis  is  ?h/-2  -    .     Hence. 

/'•e  moment  of  the  efferlice  forces  of  all  the  part iclen  of  the 
i,:ity  about  the  a.ris  is 


2.  mr 


■i 


lit'' 


(i) 


irhirli   is  the  moiiwiit  of  iiurtia  of  the  tioilij  abintt  the  a.ris 
iiiiilli/ilifil  III/  the  ani/ular  acre/era/ ion. 


I'alletl  ul^ii  Aiuiular  Momentum.    (Sie  I'irieS  Uigid  Uyuamic!',  \>.  41.) 


1 


at  a  Fixed 

5S.— Let  iiny 
aiul  lixed  in 

)i  1)0  tlu'  lua-ss 
nun  iIr'  axis, 
axis  and  tlic 


rjuMuliciilar  to 

(II ' 
larticlo    about 


}  axis  IS  /• 


le  momenta  of 

(1) 

dly  connected, 

cle,,  and  is  the 

moinoit  of  till' 

III!'  (txin  is  flu' 

s  >ni(Uij)lk'd  bij 

lie  direction  in 
tlie  moment  of 

/•2  ,^.  .     Hence. 
(ir 

par/ id  en  of  the 
y  ((hunt  till'  (i.ris 

Jyuuiiiics,  p.  41.> 


i 

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► 

7 


<9 


/} 


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ROTATlOy  OF  A    HKUD  IIODl'. 


455 


(1)  Lei  Hu'  forces  bo  impulsive  (Art.  :30'^)  ;  let  o),  o/,  he 
the  angular  velutilieri  just  liel'ore  and  just  after  the  action 
of  the  forces,  and  .V  tlie  moment  of  t lie  impressed  forces 
about  the  axis  of  rotation,  by  which  the  motion  is  pro- 
duced. 

Then,  since  by  D'Alembert's  principle  the  efreclive 
forces  when  re\ersed  are  in  e([uilibriuni  with  the  impressed 
forces,  we  have  from  (1) 

0)'  2  mi'^  —  w  i:  mr'^  =  JV"; 


w  —  w 


i;  nir^ 


moment,  of  imi)ulse  about,  axis 

moment  of  inertia    al)out.  axis' 


(3) 


that  is,  f/ic  chaiifie  in  the  aminlar  rvlorily  nf  a  hodif,  pro- 
duced bji  an  impulse,  is  etjuiil  to  the  nmuienl  of  the  iiujmlse 
divided  bij  the  moiiient  of  inertia  of  the  tjodi/. 

(2)   Let  the    forces   be    Unite.      Then    taking   moments 
about  the  a.vis  us  before,  we  have  from  (2) 


(PO 
dt^ 


1  mr^ 


moment  of  forces  al)out  axis 
moment  of  inertia  alxiut  axis  ' 


(■^) 


that  is,  the  ani/utar  amteration  of  a  liodji,  produced  hif  a 
force,  is  equal  to  the  niouie/it  of  the  force  dicided  hji  ttie 
moment  of  inertia  of  tite  body. 

\\y  integrating  (4)  we  shall  know  the  angle  Ihrougli 
which  (!•;'  Ijody  has  revolved  in  u  given  time,  'J'wo  arbi- 
li'ary  conslanis  will  .-ippear  in  llie  integrations,  whose 
\aliies  are  lo   be  delennined   from   I  he  given  initial  valuer 

of  W  and       •     Thus  (he  whole  motion  can   bo  found,  and 


456 


EXAMPLE. 


we  shall  consequently  be  able  to  determine  the  position  of 
the  body  at  any  instant. 

ScH. — It  a|)poars  from  (3)  and  (4)  that  the  motion 
of  a  rigid  body  round  a  fixed  axis,  under  the  action  of  any 
forces,  (U'lR'nds  on  (1)  the  moment  of  the  forces  about 
that  axis,  and  {'i)  the  moment  of  inertia  of  the  bod}  about 
the  axis.  If  the  wliole  mass  of  the  body  were  concentrated 
into  its  centre  of  gyration  (Art.  tiij),  and  attached  to  the 
lixed  axis  of  rotation  by  a  rod  without  mass,  whose  lengtli 
is  tlie  radius  of  gyration,  and  if  this  system  were  acted  on 
by  forces  having  the  same  moment  as  befoi-e,  and  were  set 
in  motion  with  the  same  initial  vahies  of  f>  and  the  angular 
velocity,  then  the  whole  subse(iuent  angular  motion  of  the 
rod  would  be  the  same  as  that  of  tiin  body.  Hence,  we  may 
say  brielly,  lliat  a  body  turning  al)out  a  fixed  axis  is 
kinetically  given  when  its  nniss  and  radius  of  gyration  are 
known. 

EXAMPLE. 

A  rough  circular  horizontal  Itoard  is  capable  of  revolving 
freely  round  a  vertical  axis  through  its  centre.  A  nuin 
walks  on  and  round  at  the  edge  of  the  board  ;  when  ho 
has  completed  the  circuit  what  will  l)e  bis  position  in 
space  ? 

Let  a  be  the  radius  of  the  board,  .1/  and  M'  the  masses 
of  the  board  and  man  resi)ectively  ;  0  and  0'  the  angles 
described  by  the  lH)ard  and  nan.  and  /''the  action  between 
the  feet  of  tiie  man  and  the  board. 

The  ecpnition  of  motion  of  the  board  by  (4)  is 


A 


Fa  =  7^/^•,2 


(Pd 
ill*' 


Since  the  action  between  the  man  and  tlie  board  is  con- 
tinually tangent  to  the  path  desriribed  liy  the  num,  the 
ei|uation  of  motion  of  tin'  nuiu  is,  by  (."))  of  Art.  'iO, 


TUM  voMJ-or.M)  I'EXDn.ni. 


45: 


he  position  of 


it  the  motion 
J  action  of  any 
e  forces  about 
the  bod}  about 
re  concentrated 
ittachcd  to  the 
i,  whose  lengtii 
1  were  acted  on 
?,  and  were  set 
nd  the  anguhir 
r  motion  of  the 
Hence,  we  may 
a  fixed  axis  is 
of  gyration  are 


l)lc  of  revolving 
■entre.  A  man 
)oard ;  when  lie 
his  position  in 

I  M'  the  masses 

lul  0'  tlie  angles 

action  between 


;4)is 


ic  biiard  is  eon- 
V  tiu'  nuiii,  liie 
'Art.  -^U, 


Eliminating  /'and  integrating  twice,  the  constant  being 
x-ero  in  botli  cases,  because  tiie  num  and  board  start  frt)ni 
rest,  we  get 


Mh\^Q  =  M'lfd'. 


(1) 


When  the  man  has  complete('  the  circuit  we  have  0  +  0' 
=  ^tt;  also  I'l'^  =  -•     .Substituting  these  in  (I)  we  get 

2nM 


0'  = 


2M'  +~M' 


which  gives  the  angle  in  space  described  by  the  man. 
If  jM  =  .)/',  this  becomes 


and 


0'  =  ^n; 
e  =  in, 


which  is  th.e  angle  in  space  described  by  the  board.     (See 
Routh's  Higid  Dynamics,  p.  (iT.) 

237.  The  Compound  Pendulum.  J  hiK/i/vmrrsohdnf 
n  fixed  /lorizon/dl  a.i is  (tried  mi  hij  i/rari/i/  nti/ij.  In  deleniiine 
l/ie  motion. 

Let  ABO  be  a  sectiim  of  the  l)ody  made  by 
the  ])lane  of  the  paper  passing  tiirough  (J, 
the  centre  of  gravity,  and  culling  the  axis 
of  rotation  peri)endicularly  at  <>.  l-et  0  = 
the  angle  whii'h  ()(i  makes  with  the  vertical 
OV  :  and  let  //  =  00,  /•,  =  the  i)riiici|>al 
liidius  of  gyration,  and  M  =  the  mass  of 
I  lie  body.  Then  by  (4)  of  Art.  :;»;j(i,  we 
have 

20 


458 


TJIE  VOMl-OrM)   rEM)UHM. 

d^d  _  Mijli  sin  6  _ 


dt' 


(jh 


Mgh  sin  Q 


the  negative  sign  being  taken  because  0  is  a  decreasing 
function  of  the  time. 

This  ecpiation  cannot  be  integrated  in  lini*^e  terms,  but 
if  the  oscillations  l)e  small,  we  may  develop  sin  d  and  reject 
•ill  powers  above  the  first,  and  (1)  will  become 

cPd (]h_ 


6. 


(2) 


Multiplying  bv  'Z  dO  and  integrating,  and  sui)posing  that 

the   body   began   to   move   when   0   was  ecpial    to   «,  {I) 
Itecomes 

d0^  _      f/h 


(«»  —  <92). 


Hence  denoting  the  time  of  a  complete  oscillation  by  T, 
we  have  


-=v- 


gh 


(;o 


which  gives  the  time  in  seconds,  when  Ji  and  ^;,  arc  meas- 
ured in  feet  and  ^  —  ;52.18. 

When  a  heavy  Ijody  vil)rates  about  a  horizontal  axis,  by 
the  force  of  gravity,  it  is  called  a  roin/winid  jioididh)!!. 

Coit.  1. — If  we  suppose  tlic  whole  mass  of  tlie  coniitound 
pendulum  to  be  concent  ruled  into  a  single  point,  inu!  liiis 
point  coinu'cted  with  tlie  axis  by  a  nuHliuin  without  weight, 
it  lieeomes  a  simple  poidiihini  (Art.  l'.»4).  Di'uotiiig  the 
ilistanee  of  the  point  of  concentration  from  the  axis  hy  /, 
we  have  for  tlie  linie  of  an  oscillation,  by  (1)  of  Art.  l'J-1, 


ofArt.22G],(l) 

is  ii  (lecroiisiiii,' 

tii^o  terms,  but 
sin  6  and  reject 
ne 

(2) 

siii)posing  tluit 
(liiiil    to   «,  (i) 


scilhition  by  T, 

(3) 

11(1  /;,  iiro  meas- 

'izontal  axis,  by 
/)rii(liilh)ii. 

r  liic  ('(iniitound 

point,  iinil   tiiis 

witbont  wcijilit. 

Denotiiifi'  lilt' 

ii  tlie  axis  liy  /, 

(1)  of  Art.  iw, 


CKM'UKS   OF   (>S('/IjLATI(}.\    AM)   .Sl'JSI'EASIOy.       45!» 

^  V    /'     ^^  "'^  point  ijo  so  clioson  tiiat  the  simple  jiendii- 

liiMi  will  iierforin  an  oscillation  in  tlie  same  time  as  tlic 
cornponnd  pendulum,  tliesc  two  expressions  for  the  time  of 
an  oscillation  must  be  equal  to  each  other,  and  we  shall 
have 

h 


1  = 


=  A+f  =  00', 


(^) 


(()'  being  the  point  of  concentration). 


t;ou.  2.— This  length  is  called  the  lon/fk  of  fhr  simple 
(■(piivalenf  prndaluin  ;  the  point  0  is  ciillcd  the  coilre  of 
.sns/ti'nsioii  ;  the  point  ()',  into  which  the  muss  of  the  com- 
pound jH'iiduhim  must  he  concentrated  so  that  it  will 
oscillate  in  the  same  time  as  before,  is  called  the  centre  of 
osrillah'on  :  and  a  line  through  the  centre  of  oscillation 
and  paralli'l  to  the  axis  of  siisiiension  is  called  an  axis  of 
osriUatioii. 

From  (4)  we  have 

{l-h)h  =  k^^i 


or 


GO'.  GO 


k,^. 


(5) 


Now  (5)  world  not  he  altered  if  the  place  of  0  and  ()' 
were  interchanged;  hence  if  O'  be  made  the  centre  of 
suspension,  then  O  will  be  the  centre  of  osc'illation.  'I'liiis 
///r  rt'ii/irs  if  osriUtiHiiii  and  (f  saspeiisiou  are  (oiirertibl. , 
mill  the  liini'  if  osrillalion  ahinil  rarli  is  lliv  sanir. 

Coii.  3.  — Putting  the  derivative  ot'  /  with  respect  to  h  in 
I  I)  e(pi,il  to  zero,  and  solving  for  //,  we  get 

h  =  ^-j, 


400 


KXAMl'LES. 


which  makes  /  a  mininmni,  aiul  therefore  makes  t  a  mini- 
miiiii.  Hence,  when  the  axis  of  suspension  passes  f/troni/h 
the  principal  centre  of  gyration  the  time  of  oscillation  is  a 
minimum. 

Rem. — The  problem  of  (letemiining  tlie  law  under  whieli  a  heavy 
body  swings  about  a  horizontal  axis  is  one  of  the  most  iniiM)rtaiit  in 
tlic  history  of  science.  A  simple  i)endulain  is  a  thing  of  theory;  out 
accurate  knowh'dge  of  the  acceleration  of  gravity  deiKJuds  therefore 
on  our  understanding  the  rigid  or  compound  jHindulum.  This  was 
the  first  problem  to  which  D'Alembert  applied  his  principle. 

The  problem  was  called  in  the  days  of  D'Aleml)ert,  the  "centre  of 
oscillation."  It  was  retiuired  to  find  if  there  were  a  point  at  which 
the  whole  mass  of  the  body  might  be  concentrated,  so  as  to  form  a 
simple  pendulum  whose  law  of  oscillation  was  the  same. 

The  [lositicm  of  tlu-  centre  of  oscillaticm  of  a  body  was  first  correctly 
determined  by  Iluyghens  and  published  at  Paris  in  1073.  As 
D'Alembert's  princiide  was  not  known  at  tliat  time,  Huyghens  liad  to 
discover  some  principle  for  himself.* 


EXAMPLES. 

1.  A  materiiil  straiglit  lino  oscilliites  about  an  axis  pcr- 
pcndicuhir  to  its  length  ;  find  tlie  length  of  the  eciuivalent 
siin|)le  pendulum. 

Let  ia  =  tlie  length  of  the  line,  iiud  //  the  disttineeof  its 
centre  of  gravity  from  tlie  point  of  suspension.     Then  since 


A-,^  =  --,  we  have  from  (4) 


I  =  h  + 


U 


(1) 


Cor.  1. — If  the  poiiil  of  suspension   heat  the  extremity 
of  tlio  line  (1)  lieconies 

I  =--  ^a; 


Uuuih'«  Itiyiil  pyimmicb,  ().  (}9. 


/■.'A'.1.WA/\S\ 


401 


lakes  t  a  niini- 
jim.sex  throiKjh       I 
uscillation  in  a 


ler  whieli  a  heavy 

nost  iniiM)rtaiit  in 

iig  of  thei>ry  ;  oiii 

dciHJuds  therefore 

lulum.     Tliirt  waH 

)rinciple. 

srt,  the  "centre  of 

a  point  at  which 
1,  so  as  to  form  a 
lanie. 

was  first  correctly 
ris  in  1073.  As 
,  Huygliens  liad  to 


out  an  axis  pcr- 
f  tile  cMiuivalent 

le  distil iic'o  of  its 
1)11.     Then  since 


(1) 


it  the  extremitv 


that  is,  Hio  length  of  the  e({iiivaient  simple  peiuluhim  is 
two-tliirds  of  tlie  length  of  the  rod. 

Cor.  'Z. — Let /(  =  ^a;  then  (1)  becomes 

I  =  ^a. 

Hence,  the  time  of  an  oscillation  is  the  same,  whether  the 
line  be  suspended  from  one  extremity,  or  from  a  point  one- 
third  of  its  length  from  the  extremity.  This  also  illustrates 
the  convertibility  of  the  centres  of  oscillation  and  of  sus- 
pension (See  Cor.  2). 

CoH.  3. — If  //  =  lOfl!,  then  (1)  becomes 

2.  A  circular  arc  oscillates  about  an  axis  through  its 
middle  jjoint  perpendicular  to  the  plane  of  the  arc.  Prove 
that  the  length  of  the  simple  equivalent  pendulum  is 
independent  of  the  length  of  the  arc,  and  is  e((ual  to  twice 
the  radius. 

From  Ex.  i,  Art.  2'.V),  we  have 

k»  =  h^  +  Ji;^  =  2(i-'^)aK 

From  Ex.  1,  Art.  78,  we  have 

,  sin  rt 

h  =  n  —  a • 

« 

Therefore  (4)  becomes 

l  =  2a^(\-     ^^    )^"(l-     ,r)  =  ^"-      *■ 


m 


4fiii 


LESOTII    OF   Tin-:   SKritXDS   rKSDILrM. 


.'}.   A  riglit  00110  oscilliitos  about  an  axis  inxssing  tlirou<rli 

its  vertex  ami  pcriu'iidieiilar  to  its  own  axis;  it  is  roi|iiiro(l 

to  lind   tlio  it'iifitli  of  the  sim|ilc  eciuivaioiit    poiKlnliiiii,  ( I ) 

wlion  //  is  tlio  altitiulo  of  ti>o  cone  and   l>  tiio  radius  id"     'lo 

baso,  and  {'I)  when  tiio  altitude  =  the  radius  of  the  base  =  //. 

Ml'  +  Zr\ 
A.s.   (1)--.^       ;  {i)h. 

That  is,  in  the  second  cone,  the  oontro  of  oscillation  is  in 
the  centre  of  the  base;  so  thai  the  times  of  oscillation  are 
eijual  for  axes  through  the  vortex  and  the  centre  of  the 
base  peri)cudicular  to  the  axis  of  the  cone. 

•t.  /V  sphere,  radius  a,  oscillates  about  an  axis  ;  find  the 
length  of  the  simple  ecpiivalent  pendulum,  (1)  when  the 
axis  is  tangent  to  the  sphere,  (2)  when  it  is  distant  10« 
from  the  centre  of  the  sphere,  and  (3)  when  it  is  distant 

-  from  the  eentre  of  the  sphere. 

J«.v.   {I)  la;  (•>)  W^h;  (:5) -V^'- 

238.  The  Length  of  the  Second's  Pendulum 
Determined  Experimentally. — The  time  of  oscillation 

of  a  compound   pendulum   depends  on  h  H-     ^     by   (4)  of 

Art.  2:57.  But  there  are  difHoultios  in  the  way  of  determin- 
ing h  and  k^.  The  centre,  (r.  can  not  be  got  at,  and,  as 
every  body  is  more  or  less  irregular  and  variable  in  density, 
A'l  cannot  bo  calculated  with  sufficient  accuracy.  These 
i|uantitios  must  therefore  bo  determined  from  experiments. 
Hi'ssol  observed  the  times  of  oscillation  about  different 
axes,  the  distances  l)etween  which  wore  very  accurately 
known.  Captain  Kater  employed  the  projierty  of  the 
convertibility  of  the  centres  of  suspension  and  oscillation 
(Art.  337,  Cor.  2),  as  follows  : 

Lot  tlie  pendulum  consist  of  nn  ordinnry  sfrniglit  Imr,  CO,  and  ii 
sinnll  weight,  m,  whicli  may  bo  cliunpcd  to  it  liy  means  of  a  sitcw, 
and  sliiftt'd  from  one  position  to  anotlior  on  the  pendulum.     At  tliu 


ij-yf. 


LKSUTll   (ih-   rUF.    S/yoXD'S  P HyDl'LifM. 


KM 


is'iwg  thn)u<rli 
it  is  ivi|iiin'(l 
[K'IhIiiIuiii,  ( I ) 
mil  ills  (pf  'if 
r  tlio  l)lisc  =  /;. 

r'^  (■-'.*. 

cilhitioii  is  ill 
oscillation  air 
C'l'IltlV   of  tlic 

ixis  ;  find  tlic 
(1)  when  tiio 
is  distant  10« 
n  it  is  distant 


W;  (:5)Vr/. 

i    Pendulum 

of  oscillation 

h 


by  (4)  of 


y  of  dctormin- 
got  at,  and,  as 
bio  in  density, 
anicy.  Those 
n  experiments, 
bout  different 
cry  aocnratcly 
ijiorty  of  the 
uid  oscillation 

t  Imr,  CO,  and  ii 
iieariH  of  a  screw, 
[iduluiii.     At  tilt' 


Q 


o 


Fig.  94 


points  (■  and  ()  in  two  tiiangiilar  aper- 
turcH,  al  till'  distanci'  I  ii|)nrt,  let  two  knife 
edfrpH  ot  liard  steel  lie  placed  imrallel  to 
earli  other,  and  at  riijlil  angles  to  tbo 
peiiduluni,  so  that  it  may  vibrate  ou  either 
)f  tlieni,  as  in  Vxg.  111.  Let  m  I. e  shifted 
•  ill  it  is  found  that  the  times  of  oscilhltion 
about  ('  and  O  are  exactly  the  same.  It 
remains  only  to  measure  ("0,  and  olwerve 
the  time  of  oscillation.  The  distance  be- 
tween the  two  i)oints  ('  and  ()  is  tlie  length 
of  the  simple  equivalent  pendulum.  This  diataneo  between  the  knifo 
edges  was  measured  by  Captain  Kater  wif'.i  the  greatest  care.  The 
mean  of  three  measurements  differed  bv  less  than  a  ten  thouanndth 
of  an  inch  from  each  of  the  separate  measurements. 

The  time  of  a  single  vibration  cannot  be  observed  directly,  because 
this  wonhl   re(iuire   the  fraction  of  a  second  of  time  as  shown  by  the 
clock,  to  be  estimated  either  by  the  eye  or   ear.     'J'he  dilHculty  jniiy 
be  overcome?  by  observing  the  timi-,  say  of  a  thousand  vibrations,  and 
thus   the   error   of   the   time    of  a   single    vibration    is  divided  by  a 
thousand.     The  labor  of  so  much  counting  uuiy  however  be  avoided 
by  the  use  of  "  the  method  of  coincidences."     The  ])endulum  is  placed 
in    front  of   a  clock    pendulum  whose   time  of   vibration    is  sliglitlv 
different.     Certain  marks  made  on   the  two  pendulums  are  observed 
by  a  telescope  at  the  lowest  jioint  of  their  arcs  of  vibraticn.     The  fuld 
of  view  is  limited  by  a  diapliragm  to  a  narrow  aperture  across  which 
the   marks   are    seen   to    |)ass.       At   each    succeeding   vibration    one 
]iendulum  follows  the  oth.'r   more   closely,   and  at  last   its  mark   is 
completely  covered  by  the  other  duiing  their  passage  across  ihe  field 
of  view  of  the  telescope.     After  a   few   vibration,-,  it  ajijjears  again 
Iireceding  the  other.     In  the  interval   from  one  disappearance  to  the 
next,  one  pendulum  has  made,  as  nearly  as  ])ossible,  one  complete 
oscillation  more  than  the  other.     In  this  manner  'yM)  half-vibrations  of 
a  clock  pendulum,  each  efjual  to  a  secotul,  were  found  to  correspond  to 
r);W  of  Captain   Kater's  pendulum.     The  raiio  of  the  times  of  vibra- 
tion of  the  pendulum  and  the  clock  pendulum  may  thus  be  calculated 
with  extreme  accuracy.     The  rate  of  going  of  the  clock  must  then  bo 
found  by  astronomical  means. 

The  time  of  vibration  thus  fouTul  will  require  several  corrections 
which  are  called  "reductions."  For  instance,  if  the  oscillation  be 
not  so  small  that  we  can  ]iut  sin  0^.0  in  Art.  237,  we  must  make  a 
reduction  to  infiniudy  small  arcs.     Auotlier  reduction  is  necessary  if 


U'<i      norrox  of  a  /tonv  hvaaw  irNcn.\srnAi.\Ki>. 


■we  wiwh  to  reduce  the  rcsiiilt  to  wlint  it  would  have  been  nt  the  levtl 
of  the  sea.  The  attraction  of  the  intervening,  luuu  may  l)e  allowed 
for  l>y  l>r.  Youiiy;'«  rule,  d'hil.  Tiaim.,  IHI'J).  We  may  thus  obtain 
the  force  of  frruvity  at  the  level  of  the  sea,  wiippotiiug  all  the  land 
above  this  level  were  cut  o'*"  and  the  sen  constrained  to  keep  its 
present  level.  .\s  the  level  of  the  sea  is  altered  by  the  attraction  of 
the  land,  tiirlher  corrections  are  still  necessary  if  we  wish  to  r-diice 
the  result  to  the  surfac  of  tliat  spheroid  which  most  nearly  repre- 
sents the  earth.  See  IJouth's  Higid  1)\  iiamics,  p.  77.  For  the  details 
of  this  experiment  the  student  is  rferred  to  the  Phil.  Trana.  for  1818, 
and  to  Vol.  X. 

239.  Motion  of  a  Body  when  Unconstrained.— If 

nil  iiiipiiLse  be  communicated  to  any  point  of  a  free  body 
111  ii  diroctioii  not  passing  tlirough  the  centre  of  gravity,  it 
will  jjroduce  both  translation  and  rotation. 

Let  P  be  the  impulse  imparted  to  ^p 
the  body  ai,  A.  At  ]?.  on  ilieoi)positc 
side  of  tlie  centre  (i,  i'  distiinee  GB  ^|— 
=  AG,  let  two  opposite  impulses  be  [^p 
applied,  each  eijual  to  i/';  they  will 
not  alter  the  effect.  Xow  if  i/' 
applied  at  A  is  combined  with  the  y 
at  H  which  acts  in  the  siiine  direction,  their  resultant  is  J', 
acting  at  U  and  in  the  same  direction,  and  this  produces 
translation  only,  'i'he  remaining  y  at  A  combined  with 
the  remaining  J/' at  B,  which  actn  in  the  o|)posite  direc- 
tion, form  a  couple  which  produces  rotation  about  the 
:-entre  G. 

Hence,  wJien  a  body  receives  nn  impnlse  in  a  direction 
vhich  does  not  pass  through  the  centre  of  (jravity,  that  centre 
will  (tsstune  a  motion  of  irntislntion  as  though  the  impulse 
were  applied  imwediafeh/  to  it;  and  the  body  will  hare  a 
motion  of  rotation  aboiU  the  centre  of  (jracily,  as  thoi/yh 
that  point  ice  re  fixed. 

240.  Centre  of  Percussion.— Axis  of  Spontaneous 
Rotation. — Let  Mv  represent  tlie  iinpul-^e  impressed  upon 


p 
Fig.95 


(A  /.VA7>. 


CKNTtiK   Of  I'Kh'frSSIUiY, 


4fi; 


een  nt  the  levtJ 
may  lie  allowed 
'nay  thus  obtain 
lug  all  the  land 
ined  to  keep  its 
the  attraction  of 
•e  wish  to  r-'liice 
lOht  nearly  repre- 
For  the  details 
Trans,  for  1818. 


[Strained. — If 
}i  a  free  body 
?  of  gravity,  it 


p 

Fig.95 

resultant  is  J', 
1  this  produces 
combined  with 
o|)i)osite  direc- 
ion    about   the 

in  a  direct  ion 
lify,  that  centre 
(jh  the  impulxr 
)thj  will  have  a 
ity,  as  thouijh 


Spontaneous 

mpres.sed  upon 


± 


the  body  (Fip.  90)  wliose  mass  is  M,  atul 

//  tlie  perpcndicuhir  distance,  (U).  from 

the  centre  of  gravity,  (/,  to  the  line  of 

action,  0/',  of  the  impulse.     The  een*" 

of  srravitv  will  assume  a  motion  of  t.-ms- 

lation  with  iiie  veloeit\  v,  in  a  direction 

/)arallel   to  that  of  the  imi)ulsive  force. 

Then    from    (3)    of    Art.   'Z'.\%,    we    have   for  the   angular 

velocity 

Mvh        vh 


Fig.96 


b)  =: 


MT^  ~  L? 


The  absolute  velocity  of  each  point  of  the  body  will  be 
comi)ounded  of  the  two  velocities  of  .ranslation  and  rota- 
tion. The  point  0,  for  example,  to  which  the  impulse  is 
applied,  has  a  velocity  of  translation.  Off,  equal  to  that  of 
the  centre  of  gravity,  and  a  velocity  of  rotation,  ab,  about 
the  centre  of  gravity;  so  that  the  velocity  of  any  ]K)int  at 
a  distance  a  from  the  centre,  G.  will  be  expressed  by 
V  ±  a<.);  the  upper  or  lower  sign  being  taken  according  as 
the  point  is,  or  is  not,  on  the  same  side  of  the  centre  of 
gravity  as  the  point  0.  Thus,  if  we  consider  the  motion  of 
the  body  for  a  very  short  interval  of  time,  the  line  OGC 
will  assume  the  position  bG'C,  the  point  0  remaining  at 
rest  during  this  interval ;  that  is,  while  the  point  C  would 
be  carried  forward  over  the  line  Cc  by  the  motion  of  trans- 
lation, it  would  be  carried  l)ackward  through  the  same 
distance  by  the  motion  of  rotation.  Hence,  since  the  abso- 
lute velocity  of  O  is  zero,  we  have 


V  —  au) 


0; 


w        h 


(1) 


and  hence  denoting  OC  by  /  we  have 


4fi(] 


AXIS    Oh-  Sl'oyr.WHOrs    llOTATtOX. 


I 


l\^ 


c^> 


Now  if  tliorc  liiid  been  ii  lixcd  axis  llimugli  C  pcrpcn- 
ilicular  to  the  i)liim'  of  motion,  I  lie  initial  motion  would 
liiivc  bocii  jn-cfisrly  the  same,  and  this  lixcd  axis  I'vidcntlv 
would  not  JKivo  riMvivcd  any  pri'ssui'f  from  the  iini)uls(.'. 

Wlu'u  a  riyid  body  rotates  a!)out  a  tixod  axis,  and  tlio 
body  can  be  so  struck  that  there  is  no  pressure  on  the  axis, 
any  ])oint  in  the  line  of  action  of  tiie  force  is  called  a  centre 
of  percussio)!. 

When  the  line  of  action  of  tlio  blow  is  given  and  the 
body  is  free  from  all  constraint,  so  that  it  is  capable  of 
translation  as  well  as  of  rotation,  the  axis  abont  which  the 
body  begins  to  turn  is  called  the  axis  of  sponftineoiis  rota- 
tion. It  obviously  coincides  with  the  position  of  the  fixed 
axis  in  the  first  case. 

CoK.  1. — From  (J)  we  have 

oh  —  GC-GO  =  k?; 

hence  the  points  0  aiid  C  are  convertible,  that  is,  ;/  the 
axis  of  rotation  he  supposed  to  pa^s  throwjh  tlte  point  0, 
ttie  centre  of  spontaneous  rotation  will  coincide  with  the  cen- 
tre of  percussion. 

Con.  '^». — From  (2)  it  follows,  by  comparison  with  (4)  of 
Art.  2;5T,  that  if  the  aais  of  spontaneous  rotation  niincides 
with  thi'  axis  of  suspension,  the  centre  of  percussion  coin- 
cides with  the  centre  of  oscillafion. 

Sen.— It  is  evident  that  if  there  be  a  fixed  obstacle  at  G. 
and  it  be  struck  by  the  liody  0(^  rotating  about  a  fixed 
axis  tlirough  (',  the  ol)stacle  will  receive  the  whole  force 
of  the  moving  body,  and  the  axis  will  not  receive  any. 
Hence  the  centre  of  percussion  also  detertnines  the  jiosition 
in  which  a  fixed  ol)stacle  must  be  placed,  on  which  if  the 
rotating  Itody  im])inges  and  is  brought  to  rest,  the  axis  of 
"otution  will  sutfer  no  pressure. 


gli  r  porpc'ii- 
iiiotion  would 
ixis  (.'videiitlv 
0  iiiii)ulst'. 
axis,  and  tlio 
I'  on  tlie  axis, 
called  a  centre 

riven  and  the 
is  capable  of 
lit  Avliich  tlie 
faneoMs  rota- 
\  of  the  fixed 


hat  is.  )/  the 
the  point  0, 
with  the  cen- 


M  witli  (4)  of 
'ion  cnincitfps 
•cKssion  coin- 


tl)stac]e  at  (K 
ibout  a  fixed 
'  M'liole  force 
receive  any, 
i  the  jiositioii 
which  if  Ihe 
;,  the  axis  of 


EXAMPLES. 


10) 


An  axis  through  tlie  centre  of  gravity,  parallel  lo  the 
axis  of  s|K)ntaneous  rotation,  i^  called  the  axis  of  iiisldtitane- 
oiis  rotation.     A  free  body  rotates  al)out  this  axis  (Art.  ■^;ill). 


EXAMPLES. 


1.  Find  the  centre  of  jjcrcussion  of  a  circular  plate  of 

radius  a  capable  of  rotating  about  an  axis  which  touches  it. 

a^ 
Here  ki^  ~.      ,  and  h  —  n.     Hence  from  (••2)  wo  have 


I 


a  + 


!''• 


2.  A  cylinder  is  capable  of  rotating  at)Out  the  diameter 

of  one  of  its  circular  ends  ;  find   the  centre   of  ])ercussion. 

Let  a  =  its  length,  and  b  =  the  radius  of  its  ba.se. 

W  +  4«2 

.•I«.s\   I  —       — 

Cm 

Hence  if  3/>^  =  2a'^,  the  centre  of  percussion  will  l)e  at 
the  end  of  the  cylinder.  If  t)  is  very  small  compared  with 
a.  /  ==  |y/  ;  thus  if  a  straiglit  rod  of  small  transverse  section 
is  lield  by  one  end  in  the  hand,  I  gives  the  point  a!  which 
it  may  Ite  struck  so  that  tlie  hand  will  receive  no  jur. 

241.  The  Principal  Radius  of  Gyration  Deter- 
mined Practically.- -Mount  the  body  upon  an  axis  not 
passing  through  the  centre  of  gravity,  and  cause  it  to 
oscillate;  from  the  number  of  oscillations  jieiformed  in  a 
aiven  time,  say  an  hour,  the  time  of  one  oscillation  is 
known.  'I'lu'ii  to  find  h,  which  is  the  distance  from  the 
axis  to  the  centre  of  gravity,  attach  a  spring  balance  to  the 
lower  end,  and  bring  the  centre  of  gravity  to  a  horizontal 
])lane  through  the  axis,  which  position  will  be  indicated  by 
the  maximum  reading  of  the  balance.  Knowing  the  maxi- 
mum reading.  A*,  of  the  balance,  the  weight,  U',  of  the 
body,  and  the  distance,  a,  from  the  axis  of  suspension  to 


4(;h 


THE   IIALLISTIC  I'K.WrLlM. 


tlie  point  of  attachnioiit.  we  luivo  from  the  jtriiiLMple  <>f 
moments,  Ra  =  117/,  {"nmi  which  //  is  found.  fjul)stitut- 
inp  in  (:?)  of  Art.  2:37,  this  vahie  of  //.  and  for  T  the  time 
of  an  osciUation.  /•,  becomes  known. 

242.  The  Ballistic  Pendulum.— An  interestiiig  ap- 
])lication  of  tlie  principles  of  tiie  comi)onnd  i)endiilum  is 
the  old  way  of  determining  tlie  velocity  of  a  bullet  or  caii- 
(m-ball.  It  is  a  matter  of  consi<lerable  importance  in  the 
Theory  of  Gunnery  to  determine  tlie  velocity  of  a  bullet  as 
it  issues  from  the  inoutli  of  a  gun.  It  was  to  determine 
tills  initial  velocity  that  Mr.  Robins  about  1743  invented 
the  IMlisfic  Penduhrm.  'Phis  consists  of  a  large  thick 
lieavy  ma.ss  of  wood,  suspended  from  a  horizdutal  axis  in 
the  shape  of  a  knife-edge,  after  the  mannoi  of  a  compound 
pendulum.  The  gun  is  so  i)laccd  that  a  ball  projected 
from  it  horizontally  strikes  this  pendulum  at  rest  at  a  cer- 
tain point,  and  gives  it  ;i  certain  angular  velocity  about  its 
axis.  The  velocity  of  the  ball  is  itself  too  .great  to  be 
measured  directly,  but  the  angular  velocity  communicated 
to  the  pendulum  may  be  made  as  small  as  we  plea.<e  by 
increasing  its  bulk.  The  arc  of  oscillation  being  meas- 
ured, the  velocity  of  the  bullet  can  be  found  by  calcu- 
lation. 

The  time,  which  the  bullet  takes  to  penetrate,  is  so  short 
that  we  nuiy  suppose  it  completed  before  the  pendulum  has 
sensiblv  moved  from  its  initial  position. 

Lot  .U  be  the  mass  of  the  jiendulum  and  ball;  in 
that  of  the  ball  ;  r  the  velocity  of  the  ball  at  the  instant  of 
imi»aet  ;  h  the  distance  of  the  centre  of  gra\ity  of  the  \)vn- 
duhim  and  ball  from  the  axis  of  suspension  ;  a  the  distance 
of  the  point  of  imi)act  from  the  axis  of  suspension  ;  (.'  the 
angular  velocity  ihw  to  the  blow  of  the  ball,  and  k  the 
radius  of  gyration  of  the  pendulum  and  ball.  Then  since 
the  initial  velocity  of  the  bullet  is  r,  its  impulse  is  measured 
by  lar,  and  therefore  from  (.'})  of  Art.  'IW  we  have  for  the 


Tiih:  hai.i.isiic  vkmxIjVM. 


W.\ 


|)rinc'ij)le   of 

Substitiit- 

T  the  time 


?rc.sting  a))- 
)L'iuliiliim  is 
ilk't  or  euii- 
iiiicc  in  the 
t'  a  bullet  as 
1)  tleterniitie 
43  invented 
lari^e  thick 
ntal  axis  in 
a  compound 
,11  projected 
est  at  a  cer- 
ty  ahont  its 
.great  to  be 
ninninicated 
,ve  please  by 
being  meas- 
id  by  calcu- 

L',  is  so  short 
'iidulum  iuis 

nd   ball ;    m 

le  instant  of 

of  tlie  pen- 

the  distance 

ision  ;  o)  the 

II,  and  k  the 

Then  since 

e  is  measured 

have  for  the 


initial  angular  velocity  generated  in  tlie  pendulum  liy  tliis 
impulse, 

Ull'll 

(1) 


men 


and   from    (1)   of   Art.  •^37   we   have  for   the  subsequent 
motion 


a^e         nil  . 

dp  =-  -  k-^  ^"'  '■ 


(^) 


Integrating,  and  observing  tiiat.  if  a  l)e  tiie  angle  tiirough 
which  the  pendulum  movt's,  we  iiave    ,,  —  ^^  when  0  —  U, 


</f 


d9 


and         :^  0  wlien  (f  =  k.  {'i)  i)ecomes 

M'   =      '         (1   —  COS  U). 

Eliminating  w  between  (1)  and  (3)  we  have 

■>Mk     ,   .     .     « 
V  =       ---  V'/fi  sin  ., , 
IIKI        ■  4 


(3) 


(4) 


from  which  v  becomes  known,  since  all  the  (|Uantities  in 
the  second  member  may  l)e  observed,  or  are  known. 

We  may  determine  «  as  follows:  At  a  point  in  the  jien- 
dulum  at  a  distance  //  from  the  axis  of  suspension,  attach 
ilie  end  of  a  tape,  an(l  let  the  rest  of  the  tape  be  wound 
tightly  round  a  reel  ;  as  the  peiiduhim  ascends,  let  a  length 
r  be  unwound  from   the  reel  ;  then  c  is  the  chord  of  the 

angle  «  to  the  radius  /i,  so  that  c  =  'Z/t  sin  ,.,  which  m  (4) 


Af/i; 


■II  HI, 


(5) 


'I'he  values  of  k  and  //  may  l)e  delermincd  as  in  Art.  '•v'41. 
Jl"  the  moudi  of  the  gun    i~  placed  near  lo  the  pemlulum. 


4,U 


liOTATloy  OF  A    HEAVY  POPV. 


tlie  value  of  c,  given  bv  {•)),  must  be  nearly  the  velocity  of 
l)rojeetion. 

The  velocity  nuiy  also  be  deterniin  .d  in  the  following 
manner:  Let  tiu' gun  l)e  attached  to  a  heavy  pendulum; 
when  the  gun  is  discharged  the  recoil  causes  the  pendulum 
to  turn  round  its  axis  and  to  oscillate  through  an  arc 
whicii  can  lie  measured  ;  and  the  velocity  of  the  bullet  can 
be  deduced  from  the  magnitude  of  this  arc.  (See  Price's 
Anal.  Mecii's,  Vol.  11,  x^.t-W.) 

Hi  nil-  tlu'  invention  of  the  Imllistic  iiendiilmn  by  Mr.  Robins  in 
174;i,  Imt  little  i)rogreHM  liad  been  made  in  tlio  true  thnry  of  military 
jn'ojcctiles.  liohUm'  N<  ir  Priiicipli'x  "f  (riiiiiiiri/  was  soon  translated 
into  several  languages,  and  Euler  added  to  his  liimslation  of  it  into 
(Jerman  an  exton.sive  commentary  :  tlio  work  of  Killer's  being  again 
translated  into  Knglish  in  1784.  The  es])eriment8  of  Hol)ins  were 
all  eonducted  with  musket  balls  of  about  an  ounce  weiglit,  but  they 
were  afterwards  conliiiued  during  several  years  by  Dr.  Hutton,  who 
used  cannon-balls  of  from  one  to  nearly  three  jKUinds  in  weight. 
Hutton  used  to  8us|)end  his  cannon  as  a  pendulum,  and  .measure  the 
angle  thnmgli  which  it  was  raised  by  the  discharge.  II itj  experi- 
ments are  still  regarded  as  some  of  the  most  trustworthy  on  smooth 
bore  guns.  See  Rout  it's  Rigid  Dynamics,  p.  !I4,  also  F".cyclopa>dia 
Britnnnica,  Art.  (lunnery. 

243.  Motion  of  a  Heavy  Body  about  a  Horizon- 
tal Axle  through  its  Centre.— Ixd  the  l)ody  be  asphcre 
whose  radius  is  Ji,  and  weiglit  IT,  and  let  a  weight  1'  be 
iittached  to  a  cord  wound  round  the  circumference  of  a 
wheel  on  tlie  same  axle,  the  radius  of  the  wheel  being  /• ; 
renuired  the  distance  passed  over  by  /'  in  t  seconds. 

From  (4)  of  Art.  •^•;}<!  we  have 


Pi'fl 


irX-,2  +  Pr' 


Multiplying  by  dt  and  integrating  twice,  we  have 


e  = 


(1) 


FXj,  APLES. 


4T] 


voiocity  of 

i  following 
piMiduliini  ; 
!  pendulum 
gh  an  ari' 
bullet  can 
See  Price's 


Ir.  Hobins  in 
•y  of  military 
1)11  trnnslatPil 
iim  "f  it  into 

bchifi;  iiffiiin 
Hiiliins  wcrp 
ijlit,  but  they 

Hiitliiii,  who 
Is  in  weight, 
.measure  the 
Hiij  fxpori- 
y  on  smooth 
N^'icycloptedia 


I  Horizon- 
be  a  spluTc 
L'ight  r  be 
L'rencc  of  a 
?1  being  r ; 
lids. 


the  constants  being  zero  in  both  integrations,  since  the  lx)dy 
iitarts  from  rest  when  t  =  {).    The  space  will  be  rO. 

EXAMPLES. 

1.  Let  the  body  be  a  si)here  whose  radius  is  3  ft.  and 
weight  500  lbs.;  let  P  be  50  lbs.,  and  the  radius  of  the 
wheel  6  ins.;  required  the  time  in  viiich  tiie  weight  P  will 
ilescend  through  50  ft.     (Take  ^  =  32.) 

Ans.  21  seconds. 

2.  Let  the  body  be  a  sphere  whose  radius  is  14  ins.  and 
weight  800  lbs.;  let  it  be  moved  by  a  weight  of  200  lbs. 
attached  to  a  cord  wound  round  a  wheel  the  radius  of 
which  is  one  foot ;  find  the  number  of  revolutions  of  tho 
sphere  in  eight  seconds.     (Take  y  =  32.)  A^tii.  51.3. 

244  Motirii  of  a  Wheel  and 
Axle  when  a  Given  Weight  1* 
Raises   a  Given   Weight    IF.— Let 

the  weights  P  and  IT  be  attached  to 
cords  wound  round  the  wheel  and  axle, 
respectively,  (Fig.  (I?)  ;  let  I\  and  r  be 
t)e  tiie  radii  of  the  wheel  and  axle,  and 
w  and  w'  their  weiglits;  required  the 
angular  distance  passed  over  in  t 
seconds. 
From  (4)  of  Art.  230,  we  have 

m  _         _    PR^ 

cm 


Fis.97 


Wr 


PRi  4-  Wi-i  -{-  u^lli  +  \w'r^ 


ffi 


lavG 


(1) 


EXAMPLE. 

Let  the  weight    /'  =  30  lbs.,  11'  =  80  lbs.,  w  =  8  lbs. 
and  w'  =  4  lbs.;  and  let   //  and  r  bo  10  ins.  and  4  ins.; 


472 


iioTioy  Aiiorr  a   vkhtical  axis. 


required  (1)  the  .space  passed  over  by  P  in  12  s^'coiids  if  it 
starts  from  rest,  and  (2)  the  tensions  7' and  T'  of  the  cords, 
supporting  P  and  W.     (Take  ff  —  32.) 

Ans.   (1)  'jr.;0  ft.;  (2)  T  =  31.28  lbs.;  T'  —  WM  lbs. 

245.  Motion  of  a  Rigid  Body 
about  a  Vertical  Axis.— Let  Ali 
be  a  vertical  axis  about  which  the 
body  C,  on  the  horizontal  arm  ED, 
revolves,  under  the  action  of  a  con- 
stant horizontal  force  /•',  applied  at 
the  extremity   E,  ])erpendicnlar  to 

ED.  Let  M  be  the  mass  of  the  body  whose  centre  is  C, 
and  r  and  h  the  distances  ED  and  CD,  respectively.  Then 
from  (4)  of  Art  23G,  we  have 


\ 


-^ 


Fig. 98 


d^  _  Fr 

dP  ~  Mjk'^ +  ¥)' 

Multiplying  by  df  and  integrating  twice,  observing  that 
the  constants  of  both  integrations  are  zero,  we  have 


e  = 


Fk-fl 


2J/(^•l^  +  h^) 
which  is  the  angular  space  passed  over  in  t  seconds. 


(li 


EXAMPLE. 

Lot  the  body  be  a  spiiere  whoso  radius  is  3  ft.,  whoso 
woigiit  is  000  lt)s.,  and  Hic  distance  of  wlioso  centre  fnmi 
tlie  axis  is  S  ft.,  and  let  /•'  br  a  force  of  W  lbs.  acting  at  the 
cud  of  an  arm  10  ft.  lung;  find  (1)  tiie  numl)er  of  revolu- 
tions which  the  body  will  make  about  the  axis  in  10 
minutes,  and  (2)  the  time  of  one  revolution.  (Take 
tj  =  32.)  Ah.s.   (I)  !K}1(;.3  ;  (2)  0.2  sees. 


2  s.'t'Oiids  if  it 
Z"  of  the  cords, 

'  =  TS.fi-i  ll)s. 


^ 


Fig.  98 

SO  centre  is  C, 
ctively.    Then 


observing  that 
ve  have 


0\ 


Bconds. 


is  3  ft.,  whose 
1S0  centre  from 
IS.  iictinj.^  iit  tlu' 
iil)er  (tf  revohi- 
;he  axis  in  10 
hition.  (Tako 
{2)  0.^  Hocg. 


f 


Fig.M 


VODi'   liOLLiya    DOWN  A.\  ISCLISED   PIjANE.        473 

246.  Body  Rolling  down  an  Inclined  Plane.— .1 

lioinui/encoits  spliere  rolls  d'lrectly  down  a  ronyJi  inclined 
plane  under  the  action  of  gravity.     Find  the  motion. 

Let  Fig.  !»'J  represent  a  section 
of  tJie  sphere  and  plane  made  by  a 
vertical  plane  passing  through  C, 
the  centre  of  the  sphere,  l^et  «  be 
the  inclination  of  the  jdane  to  the 
horizon,  a  the  radius  of  the  sphere,  / 
0  the  point  of  the  plane  which 
was  initially  touched  l)y  the  sphere 

at  the  point  A,  P  the  point  of  contact  at  the  time  f. 
AC'P  =  0,  which  is  the  angle  turned  througli  by  the 
spliere,  m  =  the  mass  of  the  sphere,  F  —  tlie  friction 
acting  upwards,  R  =  the  pressure  of  the  sphere  on  the 
])lane.  Then  it  is  convenient  to  choose  O  for  origin  and 
OB  for  the  axis  of  a; :  hence  OP  =  x. 

The  forces  which  act  on  the  s])here  are  (1)  the  reaction, 
7i',  ])erpendicular  to  OB  at  P.  {'i)  the  friction,  F,  acting  at 
P  along  PO,  and  (.'5)  its  weight,  nii/,  acting  vertically  at  il 
the  centre.  Now  C  evidently  moves  along  a  straight  line 
])arallel  to  the  plane;  so  tliat  for  its  motion  of  translation 
we  have,  by  resolving  along  the  plane. 


F. 


(1) 


The  spliere  evidently  rotates  about  its  point  of  contact 
with  the  plane;  but  it  may  be  considered  as  rotating  at 
any  instant  about  its  centre  ('  as  fixed;  and  the  angular 
velocity  of  (J  at  tiuit  instant  in  reference  to  P  is  the  same 
as  that  of  P  in  reference  to  V.  From  (4)  of  Art.  236,  we 
have  for  the  motion  of  rotation 


mk.^^j,  =  Fa 


(a) 


4;4        BODY  ROLLING   DOWN  AS   INCLINED  PLANE. 


and  since  the  plane  is  perfectly  rough,  so  that  the  sphere 
does  not  slide,  we  have 

X  =  .'0.  (3) 

Multiplying  (1)  by  a  and  adding  the  result  tt>  {'i),  we  get 


ma  -t:::  +  inh\* 


df^ 


dp 


m(i(j  sm  «. 


(-t) 


Differentiating    (-i)    twice   we    get  -.^  =  a  ,.^,    which 


united  to  (4)  gives 


dp 


dp' 


(Px 
dP 


«^-+T?^^"'"- 


(5) 


Since  the  sphere  is  homogeneous,  /(:,2  =  ^a?,  and  (5) 
becomes 

d'^x        .      .  .,,, 

7/72  =  y  ^'»  «  (*^) 

■/^7n'c//  f/Zms  ///('  nca'lprnfioii  down  Hip  plane. 

if  the  si)here  had  been  sliding  down  a  smooth  plane,  the 
acceleration  would  have  been  g  sin  «  (Art.  144)  ;  so  tiiat 
two-sevenths  of  gravity  is  used  in  turning  the  s})here,  and 
live-sevenths  in  urging  the  si)here  down  the  plane. 

Integrating  ((i)  twice,  and  supi)osing  the  sphere  to  start 
from  rest,  we  have 

X  =  -^n  •  sin  a  •  P 

which  (jives  the  xpace  passed  over  in  fhc  iime  t. 
Resolving  perpendicular  to  the  plane,  we  have 

R   =    VKJ  cos  t(. 

CoH. — If  the  rolling  body  were  a  circular  cylinder  with 
its  axis  horizontal,  then  /•,'-  =  hi^.  and  (5)  becomes 


fPx 
^=|i/sin«; 


(') 


'LANE. 

iit  Uie  sphoro 
t)  {:>),  wc  get 

(5) 
frt^,  and   (5) 


///  piano,  the 

44)  ;  so  tliat 

L'  sj)ht'ro,  and 

ane. 

here  to  start 


nrpiLsnt;  force. 


475 


\o 


[\ylindcr  witli 
'conies 


(7) 


so  tliat  one-third  of  gravity  is  used  in  turning  the  cylinder, 
and  two-thirds  in  urging  it  down  the  i)lune. 
From  (7)  we  have 


.r  =  \y  sin  a  •  fi 


which  (jivex  the  space  passed  over  in  the  time  t  from  rest. 


(8) 


247.  Motion  of  a  Falling  Body  under  the  Action 
of  an  Impulsive  Force  not  Directly  through  its 
Centre. — A  string  is  wound  round  tlie  circumference  of  a 
reel,  and  the  free  end  is  attached  to  a  fixed  point.  The  reel 
is  tlien  tifted  up  and  let  fall  so  that  at  the  moment  wlien 
the  string  becomes  tight  it  is  vertical  and  tangent  to  f/ie  reel 
The  whole  "'dion  being  supposed  to  ta  he  place  in  one  plane, 
determine  the  effect  of  the  impulse. 

The  reel  at  first  will  fall  vertically  without  rotation. 
Let  V  1)0  the  velocity  of  the  centre  at  the  i  lomont  when  the 
string  becomes  tight;  v' ,  w  the  velocity  of  the  centre  and 
tlie  angular  velocity  just  after  the  impulse ;  T  the  imi)ul- 
sive  tension  ;  m  tlie  mass  of  the  reel,  and  a  its  radius. 

Just  after  the  impact  tiie  part  of  the  reel  in  contact  with 
tlie  string  lias  no  velocity,  and  at  tills  instant  the  reel 
rotates  about  this  part;  l)ut  it  may  lie  considered  as 
rotating  about  its  axis  as  fixed,  and  the  angular  velocity  of 
its  axis,  at  this  instant,  in  reference  to  the  part  in  contact 
is  the  same  as  that  of  the  latter  in  reference  to  the  former. 
The  impulsive  tension  is 


T  -  m  {v  -  v'). 


(1) 


Hence  from   (;!)  of  .\ii.  'l'M\,  we  have  for  the  motion  of 
rotation 


niki^u)  =  m  {c  —  v')a. 


(2) 


4T(5 


i'.Vi' /  hSl  I  A'   FOnCE. 


Sinw  tlie  i)arl  of  tlif  ivol  in  cuiitai't  wilh  tlic  string  lui.s 
no  velocity  at  tiie  instant  of  impact,  wo  have 


V  =  ad). 


Solving  (2)  and  (3)  vvc  have 


(TV 


n^  +  k,^ 


(3) 


(4) 


If  the  reel  he  a  homogeneous  cylinder,  ^•J^  =  -,  and  we 


have  from  (3)  and  (4) 


=  !,,    '-'^I's 


(5) 


and  from  (1)  we  have  for  the  imj)ulsive  tension, 

T  =  i»n'. 

Cor. — To  find  fhe  mibsequrn/  motion.  The  centre  of  the 
reel  Jp«;('m.v  to  descend  vertically ;  and  as  there  is  no  hori- 
zontal force  on  it.  it  will  continue  to  descend  in  a  vertical 
straight  line,  and  throughout  all  its  sul)se(|uent  motion  the 
string  will  he  vertical.  The  motion  may  therefore  he 
easily  investigated,  as  in  Art.  240,  since  it  is  similar  to  the 
case  of  a  body  rolling  down  an  inclined  plane  which  is 
vertical,  the  tension  of  the  string  taking  the  place  of  the 

T 

friction    along    the    plane.      Hence    putting    «  =  ^, ,    and 

letting  the  friction  F  =z  the  tinite  tension  of  the  string,  we 
have,  from  (1)  and  (7)  of  Art.  340. 


F  =  \v)(i,     and 


(^3- 
(W 


\0 


that  is,  the  finite  tension  of  the  string  is  one-third  of  the 


EXAMPLES. 


tlic  string  luLs 


(3) 


(4) 


=  --,  and  we 

St 


(5) 


>n. 


i  centre  of  the 
ro  is  no  liori- 
l  in  a  vertical 
nt  motion  tiie 
tlierefore  he 
similar  to  the 
)lane  which  is 
e  place  of  the 


a  =  ^j ,    and 


the  string,  we 


e-third  of  the 


Wfii<4ht,  •.mil  the  reel  tlescends  with  a  uniform  aeeeleration 

off//. 

Since  the  initial  velocity  of  the  reel  from  (o)  i.s  \r,  we 
have,  for  the  space  descended  in  the  time  t  after  the  impact, 
from  (8)  of  Art.  Uij, 

X  =1  \vt-\-  y/'\     (See  Eolith's  Rigid  Dynamics,  p.  131.) 

EXAMPLES. 

1.  A  thill  rod  of  steel  10  ft.  long,  oscillates  abonf  an  axis 
passing  through  one  end  of  it ;  find  (1)  the  time  of  an 
oscillation,  and  (2)  the  number  of  oscillations  it  makes  in  a 
(lay.  Alls.   (1)  1.4;}4sec.;  {2}  m25i. 

2.  A  pendulum  oscillates  about  an  axis  ))assing  through 
its  end  ;  it  consists  of  a  steel  rod  GO  ins.  long,  with  a  rect- 
angular section  |  l)y  J  of  an  inch ;  on  this  rod  is  a  steel 
cylinder  2  in.  in  diameter  and  4  in.  long;  when  the  ends  of 
the  rod  and  cylinder  are  set  square,  find  the  time  of  an 
oscillation.  Ans.   1.1 74  sees. 

3.  Determine  the  radius  of  gyration  with  reference  to 
the  axis  of  sus})ension  of  a  body  that  makes  73  oscillations 
in  3  minutes,  the  distance  of  the  centre  of  gravity  from  tlie 
axis  being  ."5  ft.  2  in.  Ans.  5.267  ft. 

4.  Determine  the  distance  between  the  centres  of  suspen- 
sion and  oscillation  of  a  body  that  oscillates  in  2J  sec. 

Ans.  20.2G4ft. 

5.  A  thin  circular  plate  oscillates  about  an  axis  passing 
through  the  circumference  :  tiiid  the  length  of  the  simple 
ecpiivalent  pendulum,  (1)  when  tiie  a.xis  touches  the  circle 
and  is  in  its  plane,  and  (2)  when  it  is  at  right  angles  to 
the  plane  of  the  circle.  Ans.   (1)  f «  ;  (2)  |</. 

tl.  A  cube  oscillates  about  one  of  its  edges;  find  the 
length  of  the  simple  equivalent  pendulum,  the  edge  being 
—  2«.  Ans.   |rt  V2. 


m 


478 


EXAMPLES. 


7.  A  j)risni,  wliohc  cross  section  is  a  s({uare,  each  side 
being  =  2c/.  and  wiiose  length  is  I,  oscillates  about  one  of 
its  upper  edges;  find  the  length  of  the  simple  equivalent 
pendulum.  Am.  |  y/iaf+^lK 

8.  An  elliptif  lamina  is  such  tiuit  when  it  swings  about 
one    latus   rectum    as   a    horizontal    axis,   tiie   other   latus 
re(!tuin   pas.-es    through    the   centre   of   oscillation;    [)rove'' 
that  the  eccentricity  is  ^. 

!J.  Tile  density  of  a  rod  varies  as  the  distance  from  one 
end  ;  find  the  axis  perpendicular  to  it  about  wiiich  the 
time  of  oscillation  is  a  minimum,  I  being  the  length  of  the 
rod. 

Am.  Tlie  distance  of  the  axis  from  tlie  centre  of  gravity 

JS  ^.  V-'- 
o 

10.  Find  the  axis  about  whicli  an  elliptic  lamina  must 
oscillate  that  the  time  of  oscillation  may  be  a  minimum. 

Am.  The  axis  must  be  parallel  to  the"  major  axis,  and 
bisect  the  semi-minor  axis. 

11.  Find  the  centre  of  percussion  of  a  cube  which  rotates 
about  an  axis  parallel  to  the  four  parallel  edges  of  the  cube, 
and  eciuidistant  from  the  two  nearer,  as  well  as  from  the 
two  farther  edges.  Let  -ia  \w  a  side  of  the  cube,  and  let  e 
be  the  distance  of  the  rotation-axis  from  its  centre  of 
gravity. 

•>ai 
Am.   I  =  r  +       ,  where  /  is  tl>e  distance  from  the  rota- 
tion-axis to  the  centre  of  percussion. 

li.  Find  the  centre  of  percussion  of  a  sphere  which 
rotates  about  an  axis  tangent  to  its  surface. 


Alls.  I 


?"• 


i;{.  Tjet  the  body  in  Art.  •>4I5.  be  a  sphere  whose  radius  is 
ir  ins.  and  weight  1200  lbs.;  let  it  be  moved  by  a  weight 
if  250  lbs.  attached  to  a  cord  wound  rt)iind  a  wheel  whose 


EXA.\fl>LES, 


479 


iiiuare,  each  side 
tes  about  one  of 
iniple  equivak'iit 

it  swings  about 

the   other   latus 

seillatioii ;    [ji-ove' 

stance  from  one 
ibont  whicli  tlie 
the  lengtli  of  tlie 

ueutrc  of  gravity 


itic  lamina  must 
e  a  minimum, 
major  axi?*,  and 

il)e  whicli  rotates 
dges  of  the  cube, 
.veil  as  from  the 
)  cube,  and  lot  e 
)m    its  centre  of 

'0  from  the  rota- 


II  sphere   wliich 


Ahs.  1 


l«- 


e  wliose  railnis  is 
ved  by  a  wi'ight 
1  a  wheel  whose 


ladius  is  15  ins.;    find  the  number  of   revolutions  of  the 
si)here  in  10  seconds.     (//  =  3-J.)  Ans.  58.77. 

14.  Let  the  body  in  Art.  'i\',i  be  a  s|)iiere  of  radius  8  ins. 
and  weight  50U  lbs.;  let  it  be  moved  l)y  a  weight  of  100  lbs. 
attached  to  a  ct)rd  wound  round  a  wheel  whose  radius  is 
I)  in.;  find  the  number  of  revolutions  of  the  sphere  in 
5  seconds.     (//  =  IJ-.'f)  J  «.s.   rW.O'J. 

15.  In  Art.  v'44,  let  the  weight  /'  =  40  lbs.,  W  —  100 
lbs.,  w  =  I'i  ll>s.,  and  «•'  =  <;  lbs.;  and  let  II  and  /•  be 
12  ins.  and  T  ins.;  recjuired  (1)  the  spac'  passed  Mver  by  P 
in  K)  sees,  if  it  starts  from  rest,  and  {i)  tLe  tensions  T  and 
T  of  the  cords  supporting  /'  and   IT.     (//  =  ''Vi\ 

Ans.   (1)  '.»2(;.5;  {'i)  T  -  49.04  lbs..  7"  =::  80.81  lbs. 

16.  In  Art.  244,  let  the  weight  /'  =  25  lljs.,  IT  =  fiO 
lbs.,  w  =  C>  lbs.,  and  w'  =  2  lbs.:  and  let  R  and  r  bo 
8  in.  and  :5  in.;  reiiuired  (1)  the  si)ace  passed  over  by  J'  in 
10  sees,  if  it  starts  from  rest,  and  (2)  the  tensions  7'  and 
7"  of  the  cords  supporting  /'and   IT.     {;/  —  :i2^.) 

Ans.   (1)  10<».!)2ft.;  (2)  T=  2;{.21t  ll)s. ;   7' =  (il.54  lbs. 

17.  In  Art.  245,  let  tlie  body  be  a  sithere  whose  radius 
is  :]  ft.,  whose  weight  is  800  lbs.,  and  the  distance  of  whose 
centre  from  the  axis  is  !)  ft. ;  and  let  /'  be  a  force  of  GO  lbs. 
acting  at  the  end  of  an  arm  12  ft.  long;  find  (1)  the  num- 
ber of  revolutions  wiiich  the  body  will  make  about  the 
axis  in  12  min.,  and  (2)  the  time  of  one  rev(jliition. 
{(/  =  32.)  Aii.y.   (1)  1 404:5. 0  ;  (2)  (1.07  sees. 

18.  Tn  Ex.  17,  let  the  radius  —  one  foo*^,  the  weight  = 
100  lbs.,  the  distance  of  centre  from  axis  =  5  ft.,  and 
F  —  25  lbs.  acting  at  end  of  arm  8  ft.  long;  find  (1)  the 
number  of  revolutions  wliiili  the  liody  will  make  about  the 
axis  in  5  min..  and  (2)  tiie  lime  of  one  re\o!utioii. 
in  -  :52f )  .1/'^.    (1)  18i:{!).0'.l  ;  (2)  2.2;i  sees. 

19.  If  the  body  in  Art.  247  be  a  homogeneous  si)here, 
the  string  being  round  the  circumi'erence  of  a  great  circle, 


m 


480 


EXAMPLES. 


find  (1)  the  angular  velocity  just  after  the  impulse,  and  {'i) 
the  impulsive  tensicn. 


5( 


20,  A  bar,  /  feet  long,  falls  vertically,  retaining  its  hori- 
zontal position  till  it  strikes  a  fixed  obstacle  at  one  (juarter 
the  length  o'  the  bar  from  the  centre  ;  find  (1)  the  angu- 
lar velocity  of  the  bar,  {-i)  the  linear  velocity  of  its  centre 
jn«r  after  the  impulse,  and  (."5)  the  impulsive  force,  the 
velocity  at  the  instant  of  the  im])ulse  being  v. 

Ans.  (1)  ^";  {•i)\i';  (3)  4m-. 

21.  A  bar,  40  ft.  long,  falls  through  a  vertical  height  of 
50  ft.,  retaining  its  horizontal  jwsition  till  one  end  strikes 
a  fixed  obstacle  60  ft.  above  the  ground  ;  find  (1)  its  angu- 
lar velocity,  (2)  the  linear  velocity  of  its  centre  just  after 
the  impulse;  (3)  the  number  of  revolutions  it  will  make 
before  reaching  the  ground,  (4)  the  whole  time  of  falling 
to  the  ground,  and  (5)  its  linear  velocity  on  reaching  the 
ground. 


75.10. 


ins.   (1)   2.12;    (2)   42.43;    (3)   0.345;    (4)   2.79;    (5) 


he  impulse,  and  {'i) 

5v 
us.   ~-  ;  (2)  imv. 

,  retiiining  its  Iiori- 
acle  at  une  (juurter 
liiid  (1)  tile  aiigu- 
locity  of  its  centre 
npulsive  force,  tlie 
ing  V. 

{'^  \»  ;  (3)  itmv. 

I  vertical  lieiglit  of 
ill  one  end  strikes 
;  find  (1)  its  angii- 
s  centre  just  after 
tions  it  will  make 
lole  time  of  falling 
y  on  reaching  the 

:5i    (4)  ^.79;    (5) 


CHAPTER    VIII. 

MOTION    OF    A    SYSTEM    OF    RIGID    nODIES   IN    SPACE. 

248.  The  Equations  of  Motion  of  a  System  of 
Rigid  Bodies  obtained  by  D'Alembert's  Principle. — 

Let  {x,  y,  z)  be  the  position  of  the  jjarticle  m  at  the  time  i 

referred  to  any  set  of  rectangular  axes  fixed  in  space,  and 

X,  V,  Z,  the  axial  components  of  the  impressed  accelera- 

iVx    (Ptl   (Pz 
ting  forces  acting  on  the  same  particle.    Then  -^,  -.'^,  jr^, 

are  the  axial  components  of  the  accelerations  of  the  parti- 
cle ;  and  by  D'Alembert's  Principle  (Art.  2:?5)  the  forces, 

'"(^-S)'  '"(^'-S)'  '"(^-S)' 

acting  on  m  together  with  similar  forces  acting  on  every 
particle  of  the  system,  are  in  e(|uilibrinm.  Hence  by  the 
principles  of  Statics  (Art.  05)  we  have  the  following  six 
equations  of  motion  : 

.,„(.r-g)  =  o.^ 


Sw  {yZ  —  z  Y) 


I.m  (z.V  —  j-Z)  —  1,1)1 1  z 


Px 


(1) 


C^) 


482 


riiAXSLATIOS   A.\D    IWTATIOX. 


By  means  of  those  t-ix  ('(juiitions  tlie  iiiotion  of  a  rigid 
body  acted  on  l)y  any  fiiiit(!  forces,  may  be  determined. 
They  lead  immediately  to  two  important  jtropositions,  one 
of  which  enables  us  to  calculate  the  motion  of  trMmlation 
of  the  body  in  space ;  and  the  other  the  motion  of  ro/alioti. 

249.  Independence  of  the  Motion  of  Translation 
of  the  Centre  of  Gravity,  and  of  Rotation  about  an 
Axis  Passing  through  it — [.et  (7.  'i/.  i)  be  the  i)osition 
of  the  centre  of  gravity  of  the  body  at  the  time  f,  referred 
to  fixed  axes,  {x,  y,  z)  the  position  of  the  particle  m  referred 
to  the  same  axes,  (.r',  y',  z')  tiie  position  of  m  referred  to  a 
system  of  axes  passinji;  through  the  centre  of  gravity  and 
parallel  to  the  fixed  a-\  s,  and  J/ the  whole  mass.     Then 

1.  x  =  'z  +  x',    y  ==  y  +  y',    z  =  i  +  z'.  (1) 

Since  the  origin  of  the  movable  system  is  at  the  centre  of 
gravity,  we  have  (Art.  59) 

^mx'  =  ^my'  =  ^mz  =  0; 


Also 


}Lmz  =  Afx,     ^my  =  My,  I.mz  =  Me; 
,(Px     „     (Pi/        „(PJi 


(a) 
(3) 

r(Pz 


^,     (Px        .,(Px     „     (Pi)        ..(Py      ^.     cPz       „. 
•     ^"^rf/^^^^'    '""(lP=^^dP'    -'"'dt^^^dP 

Substituting  these  values  in  (1)  of  Art.  248,  wo  have 
(Px 


(4. 


M 


(Pi 
dP 


i: .  mZ. 


maam 


I  770.V. 

-'  iiiotion  of  a  rigid 
ay  be  determined, 
t  propositions,  one 
tion  of  translation 
iiKJt  ion  of  rotation. 

n  of  Translation 
station  about  an 

i)  be  the  i)osition 
lie  time  /,  referred 
jiarticle  m  referred 
if  in  referred  to  a 
tre  of  gravity  and 
le  mass.     Tlien 


I 


MOTWy    OF  A    liODY. 


483 


-I  +  z'. 

(I) 

is  at  tJie 

centre  of 

0; 

(^) 

=  0. 

(3) 

=  m-, 

248,  wo 

liavo 

(4. 


These  three  cqtiations  do  not  eontain  tiie  co-ordinates  of 
the  point  of  application  of  the  forces,  and  are  the  same  as 
those  which  would  be  obtained  for  the  motion  of  the 
ci'utre  of  gravity  supposing  the  forces  all  aj»j)lied  at  that 
point.     Hence 

ITIie  nwtimi  of  llie  centre  of  (jrnrily  of  a  system  acted  on 
/ty  any  forces  is  the  same  as  if  all  tlie  mass  were  collected  at 
I    ttie  centre  of  gravity  and  all  the  forces  loere  ajjplied  at  that 
point  parallel  tu  their  former  directions. 

"Z.  Differenhiating  (1 )  twiee  we  liavo 

(fx  _d^       dlr       (Py  _  d^J,       (Py' 
dt^  ~  df^  "^  dp '     S   ~  dP  "^  dp ' 

iPz  _  d^k      tPz' 
dp  ~  f//a+  dty 

Substituting  these  values  in  the  first  of  ecpnitions  (3)  ol 
Art.  248,  we  have 

Lm[{i  +  y')Z-(-.  +  z')r] 
I'erforming  the  operations  indicated  we  get 


=  0. 


yl-?M^-; 


yL-m 


dh' 
dp 


+  ^»./(^-'^0 


fy' 


4H4 


tra\sIjAtiox  axd  rotation. 


Omitting  the  1st,  iJd,  4th,  5th,  Cth,  and  8th  terms  which 
vanisli  by  reason  of  (2),  (3),  and  (4),  we  have 


}Lm{ii'Z-z'Y), 


Im  Ix 


dfi 


y 


j  =  ^m{x'Y—  y'X). 


similarly  from  the  other  two  etj nations  of  (2)  we  havci' 

/    tPx'  dh'\  I    ^^ 


These  three  equations  do  not  eontain  the  co-nrdinates  of 
tile  centre  of  gravity,  and  are  exactly  the  e(jnati()ns  we 
would  have  obtained  if  we  had  regarded  the  cent'-e  of 
gravity  as  a  fixed  point,  and  taken  it  as  the  origin  of 
moments.     Hence 

The  motion  of  a  bod//,  acted  on  f>y  any  fnrcex,  atmut  its 
centre  of  (jravity  in  t/ie  same  as  if  tliv  centre  (f  (jravity  were 
fixed  and  the  same  forces  acted  on  the  body.  T\\i\t  is,  from 
(4)  the  motion  of  Iraiislation  of  the  centre  of  fj rarity  of  the 
body  is  independent  of  its  rotation  ;  and  from  (5)  the  rota- 
tion of  the  body  is  independent  of  the  translatiou  of  its 
centre. 

These  two  important  propositions  are  called  respectively, 
the  principles  of  the  conservation  of  tfie  ^notions  of  ti'ansla- 
tion  and  rotation. 

Son. — By  the  first  principle  the  problem  of  finding  the 
motion  of  the  centre  of  gravity  of  a  system,  however  com- 
plex the  system  nniv  \n\  is  reduced  to  the  jiroblcm  of 
finding  the  motion  of  a  single  particle.  By  the  second 
principle  the  problem  of  fitiding  the  angular  moti(m  of  a 
free  body  in  space  is  reiliiei  d  to  that  of  determining  tho 
motion  of  that  body  alxiut  a  fixed  point. 


voASh'in'ATrn\  or  i'/.:\ri:E  f>F  huavity. 


4ftr. 


ATION. 

id  8th  terms  which 
e  have 


-y'^n 


tlie  co-urdiniites  of 

the  cqiiiitions   we 

xlod   the  cent»-e  of 

it  as  tlic  origin  of 

hy  forms,  about  its 
tre  of  gravity  were 
(hj.  That  is,  from 
're  of  fj rarity  of  the 
i  from  (5)  the  rota- 
translation  of  its 

•allod  respectively, 
notions  of  transla- 


I'm  of  finding  the 
em,  however  com- 
()  tlie  problem  of 
Hy  the  second 
giilar  motion  of  a 
f  determining  the 


Rem. — In  nsing  the  iirst  principle  it  should  be  noticed 
that  the  inijn-essed  forces  are  to  l)e  applied  at  the  centre  of 
gravity /Jrtrrt//''/  tu  their  former  directions.  Thus,  if  a  rigid 
body  be  moving  under  the  inliuence  of  a  central  force,  the 
motion  of  the  centre  of  gravity  is  not  generally  tlie  same 
as  if  the  whole  mass  Were  collected  at  the  centre  of  gravity 
and  it  were  then  acted  on  by  the  same  central  force.  What 
the  principle  asserts  is,  that,  if  the  attraction  of  the  central 
force  on  each  element  of  the  body  be  found,  the  motion  of 
the  centre  of  gravity  is  the  same  as  if  these  forces  were 
applied  at  the  centre  of  gravity  parallel  to  their  original 
directions. 

250.  The  Principle  of  the  Conservation  of  the 
Centre  of  Gravity. — Su])pose  that  a  material  system  is 
acted  on  by  no  other  forces  than  the  mutual  attractions  of 
its  parts ;  then  the  impressed  acc;elerating  forces  are  zero, 
which  give 


therefore  from  (4)  of  Art.  24!), 

we 

got 

d^l 
dt^  ~ 

"'    dt:^  - 

:0, 

(Pi 

df 

= 

di 

• '  •     dt~ 

?'q  COS  «, 

dt-^° 

cos|3, 

(Cz 

dl 

?'o  COS  y.     (1) 


where  i\  is  the  velocity  of  the  centre  of  gravity  when 
/  =  0,  and  «,  0,  ->,  are  the  angles  which  its  direction  makes 
with  the  axes.  Therefore,  calling  r  the  velocity  of  the 
centre  of  gravity  at  the  time  /,  we  have 


v  =  sj 


d^  +  dy^  +  d?  _ 
~dP  -  "«' 


(2) 


vdiich  is  midently  eonst  :nl. 


m 


(3) 


"i^^'  COXSEinWTlOX   OF  AKKAS*. 

Uvg  =  0,  the  centre  of  gravity  remains  at  rest. 
Integrating  (1)  wo  get 

X  =  v^t  cos  «  +  ff,    y  =  v^t  cos  (i  +  b, 

i  =  Vgt  cos  y  +  c ; 

ic  —  a  _y  —  b  _z  —  c 

cos  «  ~  cos  /i  ~~  cosy 

{a,  b,  c)  being  the  place  of  tlie  centre  of  gravity  of  the 
system  wlien  /  =  0.  As  (;{)  are  the  equations  of  a'straight 
line  it  follows  that  the  motion  of  t/ie  centre  of  gravity  is 
rectilinear. 

Hence  token  a  material  system  is  in  motion  under  the 
action  of  forces,  none  of  which  are  external  to  the  .system, 
then  the  centre  of  gravity  moves  uniformly  in  a  straight  line 
or  remains  at  rest. 

Rem. — Thns  the  motion  of  the  centre  of  gravity  of  a 
system  of  particles  is  not  altered  hy  their  mutual  collision, 
whatever  ilegree  of  elasticity  they  may  have,  because  a 
reaction  always  exists  equal  and  opposite  to  the  action.  If 
an  explosion  occurs  in  a  moving  body,  whereby  it  is  broken 
into  pieces,  the  line  of  motion  and  the  velocity  of  (!<e 
centre  of  gravity  of  the  body  are  not  changed  by  the 
explosion  ;  thus  the  motion  of  the  centre  of  gravity  of  the 
earth  is  unaltered  by  eartlKpiakes  ;  volcanic  explosions  on 
tile  moon  will  not  change  its  motion  in  space.  The  motion 
of  the  centre  of  gravity  of  the  solar  system  is  not  affected 
by  the  mutual  and  recii)rocal  actit)n  of  its  several  members; 
it  is  changed  only  by  the  action  of  forces  external  to  the 
system. 

251.  The  Principle  of  the  Conservation  of  Areas.— 

If  .r.  y  be  the  rectangular,  and  r,  0  the  polar  co-ordinates 
of  a  particle,  we  have 


■:as\ 

>is  at  rest. 

L'OS  (i  +  1, 


—  c 
)8  y 


(3) 


of  gravity  of  the 
atioiis  of  a  straigljt 
'enfre  of  gravity  is 

motion  under  the 
'rnal  to  tlie  system, 
y  in  a  struigtil  line 

re  of  gravity  of  a 
I'  mutual  collision, 
have,  because  a 
to  the  action.  If 
hereby  it  is  broken 
10  velocity  of  (!'o 
t  changed  by  the 
e  of  gravity  of  the 
;anic  explosions  on 
Kice.  The  motion 
•m  is  not  affected 
j  several  members; 
;es  external  to  the 


ition  of  Areas.— 

Jtolar  co-ordinates 


COASEIi  VATKi  .V   O  /••  .1  /.'  EA  S. 


^dt       y  dt  -""  dt\xl 


m 


d 


dd 


=  r^  cos^  0  -  (tan  fi)  =  r'   ,^ 
dt  dt 


(1) 


Now  \r'^dd  is  the  elementary  area  described  round  the 
origin  in  the  time  dt  by  the  projection  of  the  radius  vector 
of  the  particle  on  the  i)laiie  of  xy,  (Art.  182.)  If  twice 
this  polar  area  be  multiplied  by  the  mass  of  the  particle, 
it  is  called  ttie  area  conserved  by  the  particle  iu  the  time  di 
iou>id  the  axis  of  z.    Hence 


2m  \x 


dy 
di 


dx\ 


is  called  the  area  conserved  bj  the  system. 

lA't  dAx,  dAy,  dAz  be  twice  the  areas  described  by  the 
])rojections  of  the  radius  vector  of  the  jiarticle  m  on  the 
planes  o^  yz,  zx,  xy,  respectively  ;  then  from  (1)  we  have 

_     /   dy  dx\  dAt 

^'''Vdt-ydt)  =  -'^'dr' 


and  diflferentiating  we  get 


Im  (i 


dt^ 


(fix 


)  = 


l/n 


dt^  ' 


(2) 


If  the  impressed  accelerating  forces  are  zero  the  first 
member  of  {'i)  is  zero,  from  (5)  of  Art.  249;  therefore  the 
second  member  is  zero.     Hence 


Xw( 


iPAi 
di'' 


=  0; 


m 


48H 


c(h\sEu-\'.\'ni)\  OF  \-is  rrvA. 


similarly 


^fii  -TT^    —  0,    I.m  ^y,"  =  0; 


df^ 


dfl 


and  therefore  by  integration 

ft,  //',  //'  being  constants. 

.-.    I.mAj,  —  M,    lmA„  =  h'f,    I.mAz  =  h"i; 

the  limits  of  integration  being  snob  that  the  areas  and  the 
time  begin  sinuiltaneonsly.  Thus,  the  snm  of  the  products 
of  the  mass  of  every  particle,  and  the  projection  of  the  area 
described  by  its  radius  vector  on  each  co-ordinate  i)lane, 
varies  as  the  time.  This  theorem  is  called  the  principle  of 
the  conservation  of  areas.    That  is, 

When  a  material  syxfrm  is  in  motion  under  the  action 
offorcex,  none  of  which  are  external  to  the  system,  then  the 
sum  of  the  products  of  tlie  mass  of  each  particle  by  the  pro- 
jection, on  any  plane,  of  the  area  described  by  the  radius 
vector  of  this  particle  measured  from  any  j.  ved  point,  varies 
as  the  time  of  motion. 

252.  Conservation  of  Vis  Viva  or  Energy.*— Let 

{x,  y,  z)  be  the  place  of  the  i)article  m  at  the  time  /,  and 
let  X,  Y,  Z  be  the  axial  components  of  the  impressed 
accelerating  forces  acting  on  the  particle,  as  in  Art.  24K. 
The  axial  components  of  the  effective  forces  acting  on  the 
same  particle  at  any  time  /  are 


"^df^' 


m 


(Py 
dt^' 


m 


(Pz 
dt»' 


If  the  effective  forces  on  all  the  particles  be   reversed, 


•  See  Art. 


A, 


(Az 

at 


lit 


z  =  7i"i ; 

10  areas  and  the 
1  of  the  products 
ction  of  the  area 
)-ordinate  jilano, 
the  principle  of 

binder  the  action 
system,  then  the 
'tide  by  the  prn- 
^d  hy  the  radius 
'•ed  point,  varies 


Energy.  "^ — Lot 
the  time  /,  and 

'  the  impressed 
as  in  Art.  24H. 

s  acting  on  the 


les  be  reversed, 


coysKinATi(i.\  I'F  \/s  nvA. 


48d 


thoy  will  ho  in  ei(iiilil)riiini  with  the  whole  group  of  ini- 
|)ressed  forces  (Art.  2:5.')).  Hence,  by  the  principle  of 
"irtual  velocities  (Art.  10+),  wo  have 

„„[(x-f;)*..(r-|?).,.  +  (^-;^,.]=o,,o 

wV.oro  dr,  6y,  6z  are  any  small  arbitrary  displacements  of 
the  ])articlo  ni  parallel  to  tlio  axes,  consistent  witli  the  con- 
nection of  the  parts  of  the  system  with  one  another  at  the 
time  /. 

Now  the  spaces  actually  described  by  the  particle  m  dur- 
ing the  instant  after  the  time  /  parallel  to  the  axes  are 
consistent  with  the  connection  of  the  parts  of  the  system 
with  each  other,  and  hence  we  may  take  the  arbitrary  dis- 
placements,  &x,   6y,  (h,    to  be  respectively   equal  to    the 

actual  displacements,  .--  (5/,    ij  6t,  -f^  6t,  of  the  particle.* 

dt        dt        dt  '■ 

Making  this  substitution,  (1)  becomes 

/(Px  dx       iPy  dy       (Pz  dz\ 
\dP  di  ^  dP  dt  +  dp  dt) 

=-"'{^"£  ^  ^^h '%)■ 

Integrating,  we  get 

Em?;3  -  i;w,t'„2  =  2Swi  /  {Xdx  +  Ydy  +  Zdz),     (2) 

vhere  v  and  i\  are  the  velocities  of  the  particle  m  at  the 
times  /  and  /„. 

The  first  member  of  (2)  is  twice  tiie  vis  viva  or  kinetic 
energy  of  the  system  ac(|uired  in  its  motion  from  the  time 
/q  to  the   time   t,  under   the  action  of  the  given  forces. 

•  That  Is,  nlihough  &«  1»  not  equal  ti)  Ux,  yet  the  ratio  of  &j:  to  dj'  iu  rqual  to  the 
ratio  of  U  to  (It. 


M> 


49(» 


cnxsKHVArios  of  vis  viva. 


Tlio  sopond  m('ml)cr  oxprcsst's  twice    tlio   v^ork   done   l>v 
tliese  forces  in  the  siiiiie  time  (Art.  18!)). 

If  I  lie  seciiiKl  iiii'iiilier  of  (t>)  he  iiii  exact  ditTereiitial  of  a 
fiiiictioii  of  r.  //.  z.  so  that  ii  e(nials  '//"(.'•,  ij,  z)  ;  then  tak- 
iiifT  tlie  (ielinite  inte^'ral  hetween  the  limits  x,  y,  z  and  u-^, 
f/g,  Zf,.  corresponding  to  /  and  Z^,  (0)  l)ecomes 

imr^  -  -»,r,^-  =  y  {x,  //,  z)  -  -^/{.r,,  y„  z,).      (3) 

Now  the  second  memher  of  (•*)  is  an  exact  differential  so 
far  as  any  particU'  m  is  acted  on  by  a  centn.l  force  whoso 
centre  is  fixed  at  {a,  b,  r),  and  which  is  a  functior.  of  the 
distance  r  between  the  centre  and  {x,  y,  z)  the  phice  of  m. 
Thus,  let  Pbe  the  central  force  —  f(r),  sayj  then 


X  = 


'fir),      Y=y^'fir) 


z  =  -"F/w. 


'fi^ix-af+iy       If  +  (2  -  cf; 
.'.    rdr  =  {x  —  a)  dx  -f  {y  ._  h)  dy  +  {z  —  c)dz\ 
.'.    m  {.Ydx  +  Ydy  +  Zd?)  =  nif(r)  dr; 

which  is  an  exact  diilerontial  ;    substituting  this  in   the 
second  member  of  (2),  it 


=  2m   r/ir)dr, 

t  o 


where  the  limits  r  and  r„  correspond  to  /  and  /„. 

Also,  the  second  member  of  (2)  is  an  exact  differential, 
so  far  as  any  two  i)articles  of  the  system  are  attracted 
towards  or  repelled  from  tsich  other  by  a  force  which  varies 
as  the  mass  of  each,  and  is  a  function  of  the  distance 
between  them.  fiCt  m  and  m'  be  any  two  particles  ;  let 
(.r,  y,  z),  (x',  y',  z')  be  their  places  at  the  time  t ;  r  their 
distance  apart ;  P  =  f(r),  the  mutual  action  of  the  unit 
mass  of  each  particle.     Tlicn   the  whole  attractive  force  of 


I-l. 

'   v\()rk   (lone   ]\y 

(lifTcrciitiiil  of  ii 
y,  z) ;  tlicMi  tak- 
i  X,  y,  z  and  ^„, 
ini's 

'o.  ^0'  ^o)-       (3) 

ct  differential  so 
iitn.l  force  whoso 
function  of  the 
)  tlio  place  of  m. 
ay;  fiien 


^  =  -"?-V(r). 


{z  —  c)  (7«; 

[r)dr; 

ing  this  ill  tlie 


coysKh-iATiox  nr  r/.s   i/r.i. 


4!ll 


nd/„. 

iwat't  differential, 
Mn  are  attracted 
irco  which  varies 
of  the  distance 
vo  particles  ;  let 
time  i ;  r  their 
tion  of  the  unit 
tractive  force  of 


jn  on  m'  is  I'm,  and  the  wliole  attractive  force  of  ///  mi  //. 
is  Pm' ;  and  we  liave 


X=,n-^ ^J',     y 


1)1  •'  -  -^  /',     Z  =  III P; 


A"  =  -m""     "  P,     y=-n,^^     •' 


x—x 
r 


/• 


Also      r2  =  (r  -  x')'  +  (.'/  -  //')-  +  (z  -  z')'- 
Therefore  for  these  two  particles,  we  have 
m  {Xdx  +  Ydij  +  ir^/z)  4-  m'  {X'lU  +   )'V///'  +  Z'dz') 

=  ^  i^  [(-^  -  ^')  ('/■'•  -  '/•'•')  +  (//  -  y')  ^'^y  - '///') 

+  (z  -  2')  {dz  -  (.y)] 

=  mm'  f  {)•)  dr; 

which  is  an  exact  differential.  The  same  reasoning  applied 
to  every  two  particles  in  tiie  system  must  lead  to  a  similar 
result ;  so  tiuit  liiuilly  the  second  member  of  (2) 


=  2m7n'  I  f{r)  dr, 


where  the  limits  /•  and  r^  correspond  to  /  and  /„,  so  thut 
the  integral  will  be  a  function  solely  of  tlie  initial  and  final 
fo-ordinates  of  the  particles  of  the  system. 

Hence,  when  a  iiialvrial  xijslrm  /.s  in  motion  nndtr  the 
action  of  forces,  none  of  which  arc  e.iicrmd  to  the  system, 
then  the  chaiif/e  of  the  ris  viva  of  the  si/stem.  in  passii'y 
from  one  position  to  another,  depends  only  on  the  tiro  posi- 
tions of  the  syste^n,  and  is  independent  of  the  path  descritml 
by  each  particle  of  the  system. 

This  tlieorem  is  called  the  principle  of  the  conservation  of 
vis  viva  or  energy. 


4!»^' 


Fiiistiri.h:  or  rrs  rn'A. 


Con.  1.  —  ff  a  system  1)0  iiiulcr  tlu-  action  nf  no  oxtcrnal 
forces,  we  liave  X  =  V  =  Z  =  (»,  and  lience  the  vis  viva 
of  tlie  system  is  constant. 

CoK.  2. — Let  gravity  he  the  only  force  acting  on  tlie 
system.  Let  the  axis  of  z  he  vertical  and  positive  down- 
wards, then  wo  have  ,r  _  (I.  r  _  0,  Z  —  :/.  llenee  (•.') 
becomes 

Itnv'^  —  ^mi-g^  =  21//J  {z  —  z„). 

But  if  z  and  z,  are  the  distances  from  tlie  plane  of  .ri/  to  the 
centre  of  gravity  of  the  system  at  the  times  /  and  /„.  and  if 
M  is  the  mass  of  the  system,  we  have 

Mi  —  I.mz,     .!/?„  =  -iiiz^; 
.'.    Smi-^  —  I.mr^^  =  ^.Vy  (5  —  ig).  (4) 

Tiiat  is,  fjte  inrtrase  of  vix  viva  of  the  si/sfrni  drpeniU  only 
OH  the  verticdl  dislance  over  which  the  centre  of  yravity 
passes  ;  dud  therefore  the  cis  vica  is  the  s(i)iie  whenever  the 
centre  of  gravity  passes  thronyh  a  yiren  horizontal  plane. 

HkM. — The  1  iple  of  vis  viva  whs  first  used  l)y  Huyghons  in 
)iis  determinati  m  of  the  centre  of  osfillatifin  of  a  body  (Art.  2:i~, 
Hi>ni. ). 

Tlie  advantage  of  this  principle  is  that  it  gives  at  once  a  relation 
between  tli"  velocities  of  the  bodies  considered  and  the  co-ordinates 
which  detenuine  their  positions  in  space,  so  tliat  when,  from  the 
nature  of  the  ])roblem,  the  position  of  all  the  bodies  may  be  made  to 
depend  on  one  variable,  thi:  e(iaation  of  vis  viva  is  sufficient  to  deter- 
mine the  motion. 

Supix)se  a  weight  mg  to  be  placed  at  any  height  h  above  the  snr- 
face  of  tlie  earth,  .^s  it  falls  through  a  height  z,  the  force  of  gravity 
does  work  which  is  measured  by  mijz.  The  weight  has  acquired  a 
velocity  r,  and  therefore  its  vis  viva  is  Iwi:'  which  is  equal  to  mgz 
(An,  217).  If  tlie  weight  falls  through  the  remainder  of  the  height 
//,  gravity  does  rnori'  work  whicli  is  measured  by  mg  (ti  —  z).  Wlien 
tlie  weight  has  reached  the  ground,  it  has  fallen  as  far  as  the  circuui 


II  I  if  no  external 
"nee  tlie  vis  viva 


e  iictinp:  (in  tlie 
I  {)()sitive  (It)Nvn- 
-  ii.     Hence  (•-') 

)hme  of  ;n/  to  the 
i  /  and  /„.  and  if 


'«)• 


(4) 


On  (h'penih  only 
cntrc  of  (jravity 
nil'  v/ienert'f  llie 
•izofifal  plane. 

?(J  by  Huyghons  in 
a  Ixidy  (Art.  2;{7, 

nt  once  n  relation 
id  tlie  CD-ordinutes 
at  when,  from  the 
=s  may  be  made  to 
1  Riifflcient  to  deter- 

it  fi  al)ove  tiic  Hiir- 
he  force  of  gravity 
gilt  has  ac(iuired  a 
ich  is  equal  to  mgz 
nder  of  the  heipjht 
Dig  (h  —  z).  Wlicii 
J  far  as  tbe  circuiu 


coMi'osiTio.y  II r  RorArioss. 


4();5 


stances  of  tlie  eaue  permit,  and  j^ruvily  tins  done  work  wliicli  is  meas- 
ured by  infill,  anil  can  do  no  more  work  until  tbe  weight  has  liei'n 
lifted  up  again,  llenee.  tbroughmii  the  motion  when  the  weight  ban 
de8ceu''ed  through  any  space  z,  its  vis  viva.  \nir-\  mgz\,  together 
with  the  Work  that  can  be  done  diiriiit;  the  rest  of  the  desient, 
nig {h  —  J),  is  c  msli.ni  sind  etpial  to  mgli,  the  work  done  by  gravity 
during  tliu  whole  descent  //. 

If  we  conipliciili'  the  motion  by  making  the  weight  work  .-lome 
niaeliiiie  during  its  descent,  the  same  theorem  is  still  true.  The  vis 
viva  of  the  weight,  when  it  has  descended  any  spai'e  z,  is  e<iual  to  the 
work  mgz  which  has  been  done  by  gravity  during  this  ('escent,  dimin- 
ished by  the  work  done  on  the  niachiiie.  Hence,  as  before,  the  vis 
viva  together  with  the  ditterence  lii'tweeii  the  work  done  by  gravity 
and  that  done  on  the  machine  during  I  hi'  remainder  of  the  descent  is 
constant  and  ('((ual  to  the  excess,  i  f  the  work  done  liy  gravity  over 
that  done  on  the  machine  during  the  whole  descent.  (Sec  liouth's 
Rigid  Dynamic!--,  ]>.  270.) 

253.  Composition  of  Rotations. —  It  i.s  often  neees- 
sarv  to  ciinijiound  fotationti  alioiit  axes  wliieli  meet  at  a 
point.  Wiien  a  body  is  said  to  liave  angular  velocities 
about  three  ditTereiit  axes  at  the  same  time,  it  i.soidy  meant 
that  the  motion  may  be  determined  a.s  follows:  Divide  the 
whole  time  into  a  number  of  infinitesimal  intervals  eaeh 
e([ual  to  ill.  During  eaeii  of  these,  turn  the  body  round 
the  three  axes  successively,  through  aughs  (.)^ill,o)^i/l,tj.^i/l. 
The  result  will  be  the  same  in  whatever  order  the  rotations 
take  place.  'I'hc  final  dLsjilacemeiit  of  the  body  is  the 
diagonal  of  the  parallelopiped  described  on  these  three  lines 
;!S  sides,  and  is  therefore  inilependent  of  the  order  of  the 
rotations.  Since  then  the  tiiree  successive  rotations  are 
(juite  independent,  they  may  be  said  to  take  place  simul- 
taneously. 

Hence  we  infer  that  anguliir  velocities  ami  anguhir  accel- 
erations may  be  cotniioundcd  and  resolveil  liy  the  same 
rules  and  in  the  same  way  as  if  they  were  linear.  Thus. an 
anguhir  velocity  (•>  aliont  any  given  axis  may  be  resohed 
into  two.  <.)  CO.-  ((  and  i-i  .in  </,  i'bout  exes  at  right  angles  to 


m 


494 


Mui'iox  Oh'  A  nidi  I)  rwDr. 


each  otiier  and  niakiiig  angles  a  and  ,^  —  «  witli  tlic  given 


axLs. 


Also,  if  a  body  liavc  angular  vcldcities  Mj,  o)g,  m^  alxint 
three  axes  at  right  angles,  they  are  together  e<|uivaleiit  to 
a  single  angular  velocity  w,  where  w  =  \/io];^  +  i.y^^M"'\ 
al)out  an  axis  inclined  to  tlie  given  axes  at  angles  wliose 

cosines  aro  respectively     ',     *,     •^. 

•       W        6)        w 

254.  Motion  of  a  Rigid  Body  referred  to  Fixed 
Axes. — Let  us  suppose  that  one  point  in  the  hody  is  (ixed. 
Let  this  point  l)e  taken  as  the  origin  of  co-ordinates,  and 
l(>t  the  axes  OX,  OV,  OZ  he  any  directions  fixed  in  spjice 
and  at  right  angles  to  one  another.  Tlic  body  at  the  time 
/  is  fiiniing  about  some  axis  of  instantaneous  rotation 
(Art.  ^40).  Let  its  angular  velocity  about  this  axis  I)e  w, 
and  let  this  be  resolved  into  the  angulai'  velocities  w,.  6)„, 
<.)3  :'.''<'iit  the  co-ordinate  axes.     It  is  ref|uii'ed  to  l.,id  the 

resolved  linear  velocities,  'f   ' f   '  j,  parallel  to  the  axes  of 

lit    III     (11 

co-ordinates,  of  a  particle  m  at  tlie  point   P,  (./•,  y,  z),  in 

terms  of  the  angular  velocities  about  the  axes. 

These  angular  velocities  arc  sup- 
posed positive  "hen  tliey  tend  the 
same  way  rnund  (he  axes  that 
positive  couples  tend  in  Statics 
(Art.  (m).  Thus  the  positive 
directions  of  f.),.  <.)„.  m.^  are  re- 
spectively from  //  to  z  about  :/•, 
froni  z  to  ./;  about  //.  and  from  r 
to  //  about  X '.  jMid  (hose  negative 
which  act  in  (lu'  oj)po.sito  direc- 
(iiins, 

liCl     M-    de(ennine   (he    V('loei(\ 
"f  /'  piiiallel  (o  the  axis  of  z.     Let    l' X  be  the  ordinate  z, 


r. 


^IA'/,V   OF  IXSTAXTAXhOrs  ROTATION. 


495 


'<  with  tlie  given        t 


',,  <.)g,  M.^  al)()iit 
er  t'((uivaleiit  lo 

at  angles  wliose 


rred  to  Fixed 

lie  l)0(ly  is  (ixod. 
o-ordiiiates,  and 
IS  fixed  in  gj)jM?e 
)()dy  at  the  time 
aneoiis  rotation 
tins  axis  lie  w, 
c'locities  6),.  oig, 
I'ed   tu  l.,id  tlio 

el  to  tlie  axes  of 

A  (•'•,  y,  z),  in 
es. 


llie  ordinate  z, 


\ 


and  draw  IWf  jierpendienlar  to  the  axis  of  x.  The  velocity 
(if /Mue  to  rotation  aliout  OX  is  m^PM.  Resolviii"-  this 
parallel  to  tlie  axes  of  //  and  z,  and  reckoning  tliose  linear 
velocities  jiositive  which  tend  from  the  origin,  and  vice 
rerun,  we  have  the  velocity 


along 


MN  =  -  w, PM  cos  XPM  =  —  oi.z 


i*» 


I        and  along       NP  ~  (i>iPM  sin  NPM  =  w,?/. 

I  Similarly  the  velocity  due  to  (he  rotation  abont  OV  par- 

I        allel  to  CXis  <.>„z,  ami  parallel  to  OZ  k  —  („„x.     And  that 
iiie  to  the  rotation  ahont  OZ parallel  to  OA' is  —  ut^y,  and 


parallel  to  01'  is  (.).,.f. 


Adding  together  those  velocities  which  are  parallel  to 
the  same  axes,  we  have  fen- the  velocities  of  P  parallel  to 
the  axes  of  x,  y,  and  z,  respectively. 


doc 

dv 

dz 

^  =  u,,y-  u,,x 


(1) 


255.  Axis  of  Instantaneous  Rotation.— Every  jiar- 

ticle  in  the  axis  of  insiaiitaneoiis  rotation  is  at  rest  relative 
to  the  origin;  hence,  for  these  particles  each  of  1  lie  first 
members  of  (1)  in  Art.  -^M,  will  reduce  to  zero,  and  w 
have 


6),2  -  6)3?/  =  0, ' 

6)3.7;  —  (i)j2   =   0, 


(1) 


490 


A^OLLAH    VELOVITr. 


which  are  the  equal  ions  of  the  axis  of  instantaneous  rota- 
lion,  the  third  equation  being  a  necessary  consequence  of 
the  first  two  j  heuoe, 


6),     '       -5'  O),      ' 


(2) 


that  is,  the  instiintaiioous  axis  i,s  a  straiglit  lino  passing 
throiigli  the  origin  wliich  is  at  rest  at  tlio  instant  con- 
sidered ;  and  the  whole  body  must,  for  the  instant,  rotate 
about  this  liue. 

Cor. — Denote  by  «,  (i,  y  the  angles  which  this  axis 
makes  with  the  co-ordinate  axes  x,  y,  z,  respectively, 
then  (Anal.  Geom.,  Art.  175)  we  have 


cos  f<  = 


w. 


Vw,«  +  Wi*  -f-  Ws» 


COS  (3  = 


Vw,2  +    Wj^  +   W3« 


COS  y  =: 


\/Wi*  -f   Wg*   +   0)3'' 


which  gives  the  position  of  the  instant (nieons  axis  in  termi* 
of  t lie  angular  rrtorities  aliout  tlie  co-ordinate  axes. 

256.  The  Angular  Velocity  of  the  Body  about  the 
Axis  of  Instantaneous  Rotation.— Tlie  angular  veloc- 
ity of  the  body  iilxmi  this  axis  will  lie  the  snnic  as  that  of 
any  single  particle  chosen  iit  pleasure.  Let  the  particle  be 
taken  on  the  axis  of  x  ;  if  i'roni  it  we  draw  a  perpendicular, 
/).  to  the  instantaiK'ous  axis,  llicn  the  distance  of  the  par- 
ticle from  the  origin  being  ./■,  we  have 


EULEIiS   EH  I  ■.  1  TWytl. 


4U? 


ttantuHvoiis  rotU' 
r  consequence  oj 


(2) 

Tht  lino  passing 
\\o  instant  con- 
e  instant,  rotutu 

\\\\\c\\   til  is  axis 
z,   respectively, 


s  fixis  in  tertm 
e  axes. 

ody  about  the 

'  angular  voloc- 
sanic  as  lliat  of 
the  particle  lie 
I  perpendieular, 
nee  of  the  par- 


p  =  xsm 


rt  =  a;  Vl  —  eos^  a  =  x\/      ~*— d— i 


Since,  for  this  particle,  ^  =  U,  «  =  0,  we  have  from  (1) 
of  Art.  'Zb\,  for  the  al)Solute  velocity, 


,^       Vdx^  -f-  dif  +  dz^  .  - 

V  = ~ =  X  yu) 


dt 


*  4-  f.)   * 
2      T^   "^S  I 


and  hence,  for  the  angular  velocity  v,  we  have 
V 


V 


-    \/w,«  +    Wjj2  -1- 


6) 


3  > 


which  is  the  am/ii/dv  velocily  required. 

257.  Euler's  Equations. —  To  determine  the  general 
equuliuns  of  motion  of  a  body  about  a  fixed  point. 

Lot  the  fixed  jioint  0  he  taken  as  origin  ;  let  {x,  y,  z)  he 
(he  plai'e  of  any  particle  in.  at  the  time  /,  referred  to  any 
rectangular  axes  lixed  in  space,  and  let  0:r,,  Oy^.  Oz^  he 
the  lu'incipal  axes  of  the  hody  (Art.  231).  Differentiating 
(1)  of  Art.  '^54  with  respect  to  /,  we  have 


<Px  dM„  dut,    ,  ,  ,  , 

"  -  .'/  -/,    +  "'2  ('■'1.'/  —  '■»2P)  —  Wj  ((.)3.r— <.), 


df^  ~  ^  dt 


z). 


dh 

dh 

dt^ 


di 

d(>> , 


du 


—  ^  It  +  "'3  ('*'2«  -  "'3^)  -  '•'i  ('^ly  —  <^«^)> 


dt 


•'/ ,//  -  -  dt 


^  .n   +  '''1  ('^3'-  -  '''i^)  -  <^«  ('^««  —  "»y)- 


Henoting  ity  /,.   I/,  .V,  tin-  lirst  terms  respectively  of  (".'). 
(An.  •-MH),  and  suhstiliiting  the  ahove  values  of   ,  ^  and     ' 
in  the  last  of  these  eipnilions,  we  get 


•198 


EULET.\S   ICqVATIO^S. 


l.m{x^+y^)  j^  —  'Lmyz 


<w„ 


dt 


-r  —i^mxz 


(=A 


(1) 


'I'he  other  two  equations  iiiay  be  treated  in  the  sunie  way. 

The  coefficients  in  this  equation  are  the  moments  wnA 
))roihicts  of  inertia  of  tiie  body  with  regard  to  axes  lixed  in 
space  (Art.  22-t),  and  are  therefore  varial)le  as  tlie  b^dy 
moves  about.  Let  w^.,  lOy,  w^  be  the  angular  velocities  about 
the  principal  axes.  fSince  tiie  axes  fixed  in  space  are  per- 
fectly arbitrary,  let  them  l)e  so  chosen  that  the  ])rincipal 
axes  are  coinciding  with  them  at  the  moment  under  con- 
sideration.    Then  at  this  moment  we  have  (Art.  '^3;^;, 

^mxy  =  0,    l.niyz  =  0,     l^mzx  =  0 ; 


also  o)j  =  (Oj.,  Wjj  =  Wj,,  Wj  =r  lOg  ;  and  likewise 


dt 


d<<>^ 
dt' 


etc*     Hence,  denoting  by  A,  B,  (\  the  moments  of  inertia 
about  the  principal  axes  (Art.  ^31),  (1)  becomes 


in  which  all  the  coefficients  are  constants;  and  similarly 
for  the  other  two  e(|uatioiis. 

Hence,  uniting  them  in  order,  and  retaining  the  letters 
a),,  Wg,  Wj,  since   they  are  e(jual  to  w.,.,  u^,  w^.  the  three 


lif  ~   lit  '  "  •■''""-''"  '"  ""'  '"■"  niitriil'ir  vrldciticH,  (u,  anil  ...,,  diiriii;;  n 

(jiv.'ii  Hiniill  time  iillcT  llic  w^W  nf  .c,  iDiiicUlfs  will,  til,,  ii.xi-  (if  ./,  will  (IlllVr  onlv  Iit 
Hi|ii.uii;iy  Hliiili   Icpciids  ii|><)ii  Ihciuiifln  piwscMl  Ihmiisri,  by  tluuixiHor.-',  dmliiK 

'timl   ulviMi   small   llnio  ;  ilm  .llir.Mvi.tv  bclw.rti  ,.,,   ai will  IIiitWoiv  bran 

int.  iKi'simiil  olth.-scnmil  oidrr  ami  Ih.Tifori'  their  dcrivallvcx  will  becciiiul.  (Scu 
I'iMllV  Mpch.,  p.  ISW.  K(.r  fnitlMT  .lemoiiMnili.ih  of  this  fipmllly,  llic  student  l« 
If I'uirod  to  UoiitU'a  lligid  Uyimmic>s  1>1>.  ISH  and  18U.) 


t 


(1) 


0  Siinio  way. 
inomonts  ;>iul 
t  axcti  lixod  in 

as  tlio  l)jdy 
olot'ities  about 
space  arc  jxt- 

tlie  ])rincii)al 
it  umlor  con- 
•t.  )i;i:ij, 


!C       ,,'    = 


(If  ~  df 

L'lits  of  inertia 

108 


luul  similarly 

g  tlio   k'ttcrs 
w«,  tlic  llircc 


ij,  anil  >.p,,  iliirili;;  ft 

,  will  (lllUTonty  liT 
I!  axis  of  .1',  diiriiiK 
III  thi'ri'lnri-  be  an 

iVill  I)C>C'(|IIU1.      (SiH! 

lly,  tlic  ctuiliiit  l« 


E  uIj  t:u  's  KQ  i:a  ttons. 


499 


equations  of  motion  oj  the  body  referred  to  the  principal 
axes  at  the  fixed  point  are 


A%'.-yn-n 


'i"'i 


=  L, 


dt 


iy-^,2  _  (6'_  j)o)3,.)j  =  M, 


(2) 


"   dl 


(-1 


B)  WjW^     =    ^> 


These  arc  called  Euler's  Ecpjations. 

Kcir. — If  the  l)ody  is  moving  so  there  is  no  point  in  it 
which  is  fixed  in  space,  the  motion  of  the  body  al)out  its 
centre  of  gravity  is  the  Hime  as  if  that  point  were  fixed. 

It  is  clear  that,  iuiUcad  of  referring  tiie  motion  of  the 
liody  to  tiic  principal  axes  at  the  fixed  point,  an  Euler  has 
done,  we  may  use  an\  axes  fixed  in  the  body.  JUit  these 
are  in  general  so  comj  Heated  as  to  be  nearly  useless. 

258.  Motion  of  a  Body  about  a  Principal  Axis 
through  its  Centre  of  Gravity. — //'  a  body  rolato  about 
our  of  its  principnl  nxt'x  passing  t/iroufffi  flic  centre  of 
(/rnviti/,  this  axis  irill  miffer  no  /treasure  from  the  centrifn- 
gnl  force. 

Let  the  body  rotate  about  the  axis  of  «;  then  if  u)  be  its 
angular  velocity,  the  centrifugal  force  of  any  jjarticle  ni 
will  be  (Art.  I'JH,  Cor.  1) 

which  gives  for  the  .i-and  //-components  w/o)'./' and  w((i)*y ; 
and  (he  moments  of  these  t'orecs  with  respect  to  (he  axes  of 
y  and  x  are  for  the  wliole  body 

l.ini>>^xz,    and     ^un^yz. 


500 


AXIS   OF  i'A/.M/.I.VA'.Vy  ROTAriOX. 


IJut  thcso  aro  "iich  ociuai  to  zero  when  tlio  axis  of  rotation 
is  a  principal  axis  (Art.  •-i;5:i) ;  iiciice,  tlie  centrifugal  force 
will  have  no  tendency  to  iiniiiic  the  axis  of  z  towards  the 
]ihuK'  of  .ry.  In  this  case  tlio  only  ctfcct  of  the  forces  miJ^x 
ami  nu^y^y  ou  the  axis  is  to  move  it  parallel  to  itself,  or  to 
1  ranslate  the 'jody  in  the  directions  of  x  and  y.  But  the 
sum  of  all  these  forces  is 

^tnui^x    and     ^nu<y^y, 


each  of  which  is  eqiud  to  zero  when  the  axis  of  rotation 
passes  through  tiie  centre  of  gravity ;  hence  we  conclude 
that,  lohen  a  butly  rotates  about  one  of  its  princiiml  (turs 
passing  t/trouyh  its  centre  of  ynirity,  the  rotation  causes  no 
pressure  upon  tin'  axis. 

If  the  l)ody  rotates  al)out  this  axis  it  will  continue  to 
rotate  about  it  if  tlie  axis  he  removed.  On  this  account  a 
principal  axis  tlirough  the  centre  of  gravity  is  called  an 
axis  of  pernuuient  rotation.* 

Sen. — If  the  l)oily  be  free,  and  it  begins  to  rotate  about 
an  axis  very  near  to  a  princii)al  axis,  the  centrifugal  force 
will  cause  tiie  axis  of  rotation  to  change  continually,  inas- 
much as  the  foregoing  conditions  cannot  obtain,  and  this 
axis  of  rotation  will  citiier  continually  oscillate  about  the 
principal  axis,  always  rcuiiiining  vei'y  near  to  it,  or  else  it 
will  remove  itself  indctinitely  from  the  jjrincipal  axis. 
Hence,  whenever  we  observe  a  free  body  rotating  about  an 
axis  during  any  lime,  however  short,  we  may  infer  tliat  it 
has  continued  to  rotate  al)out  tiiat  axis  from  tiie  beginning 
of  the  motion,  and  that  it  will  continue  to  rotate  about  it 
forever,  unless  checked  bv  some  extraneous  obstacle.  (See 
Young's  Meehs..  p.  "•:{().  also  Venturoli.  pp.  K?")  and  KiO.) 


*  Pralt"«  MccliH..  |).  ia.>.    Ciillcil  al<c)  a  nalnnil  n.ii«  ql'  ifilatinn.  hcc  Y<)iiii(t'ii 
MechH..  p.  830  ;  alw  an  inrariablc  «.Ws,  fi't'  I'liccV  Mi'cli^.,  Vol.  II,  I'  4''7. 


VOA-. 


VELOCITY  ABOUT  A   PRISCII'AL  AXIS. 


OUI 


axis  of  rotation 
centrifugal  force 
of  z  toM'ards  tlie 
'  the  forces  mul^x 
el  to  it.self,  or  to 
md  y.     But  the 


axis  of  rotation 
CO  we  conclude 
?  priuciiHtl  axes 
'at ion  causes  no 

ivill  continue  to 

tiiis  account  a 

ity  is  called  an 


to  rotate  about 
'ntrifugal  force 
mtinually,  inas- 
btaiu,  and  this 
illato  about  flic 
to  it,  or  else  it 

l)rincii)al  axis, 
ating  alioMt  an 
v  infer   that   it 

the  bcginnijiif 
rotate  alxuit   it 

obstacle.  (See 
i:)'}  and  1(10.) 


tffi/inii,  Kcc   VdiiUK's 
1.  II,  p.  *'<7. 


259.  Velocity  about  a  Principal  Axis  when  there 
are  no  Accelerating  Forces. — In  liiis  case  L  =  .V  = 
y  =  0  in  (2)  of  Art.  :^57  ;  also  A,  B,  C  are  constant  for 
the  same  bjdy;  and  if  we  put 

B-C  ,,      C-A  ^       A-  B  rr 

~A~  =  ^'    ~B~  =  ^'    ~C~  =  ^' 

{'■I)  of  Art.  257  becomes 

du^  =  Huy^io^dt. 

Put  WjWgWg^/^  =  dtp,  and  we  have  (1) 

6)jr?Wi  =  Fdcp,    Wg</wj  =  Gd(p,    M^dio^  =  Hd(f>\ 

and  integrating,  we  get 

w,»  =  -IF^  +  a\  wg"  =  'iG(p  +  b\  Wj^  =  UI<t>  +  <^.  (2) 

where  a,  b,  c  are  the  initial  values  of  Wj,  w^,  Wj  ;*  hence 
from  (1)  and  (2) 


dt  — 


V(2i^  +  o?)  (^G^b  +  A^)  (2H(p  +  (?) 


(3) 


Suppose  now  the  body  begins  to  turn  about  only  one  of 
tlie  principal  axes,  say  the  axis  of  ,r,  with  the  angular 
velocity  a,  then  J  =  0,  c  =  0,  and  (3)  becomes 


dl  = 


d<t> 


2VGH  (fiV^iFtp  +  a« 

Replacing  '2F<f>  -f  «'  by  its  value  Wj^,  and  d<p  by  its  value 

u>,dM.         , 
-n—>  we  have 
F 

dt  = 


1 


dw^ 


V'Gir  «•>,'-«*' 


ita 


oOi  THE  lyTEGUAL    OF  ErhEli'S  EQl'ATIOSS. 

and  integrating,  we  get 

2«     "  Wj  +  a 


Wj  -f-  a 


(4) 


Die  constant  C  must  he  determined  so  that  when  /  =  0. 
(')y  is  tlie  initial  velocity  a;  hence  eP^  =  0  or  C  =  —  go, 
which  makes  the  first  member  of  (4:)  zero  for  every  value 
of  /.  Hence,  at  any  time  /,  we  must  have  w,  =  «;  and 
therefore  from  (2)  <f>  =  0,  and  Wj  =  Wj  r=  0.  Conxe- 
qiiciitlji  the  impressed  velorify  about  one  of  the  principal 
(i.res  of  rotation  continnes  perpetual  and  uniform,  as  before 
shown  (Art.  258). 

260.    The  Integral  of  Euler's  Equations. — A  body 

rernires  abonl  its  centre  of  ijravitif  acted  on  by  no  forces  but 
snc/i  as  pass  througti  that  point  ;  to  intetj rate.  Ike  equations 
of  mat  ion. 

As  the  only  forces  acting  on  the  body  are  tliose  which 
pass  through  its  centre  of  gravity,  they  create  no  moment 
(if  rotation  about  an  axis  passing  through  that  centre;  and 
therefore  (2)  of  Art.  257  become 


^'^ 

-(/>'-   (7)6,20)3 

=  0, 

^'s« 

—  (C —  A)  WjW, 

=  0, 

(It 

—  {A  —  B)  <.),a)2 

=  0, 

(1) 


t'lC   priiu'ipal    axes    Ijeing   drawn    through    the   centre   of 
gravity. 


Tioys. 


(4) 

it  when  /  =  0. 
or  6'  =  —  00 , 
for  every  value 
■  Wj  =  «;  and 
—  0.  Conse- 
''  flie  principal 
form,  ua  before 


Jons. — A  body 
'I  iiu  furciKi  (ml 
<}.  the  equations 

re  tliose  which 
ite  no  moment 
lat  centre ;  and 


(1) 


tlie   centre   <»f 


r 


THE  INTEHRAL    OF  ECLKIt's   EQUATIONS. 


503 


Multiply  these  erjuiitiotis  severally  (1)  by  w,,  w,,  w,  ; 
and  {2)  by  Jwj,  Bo)^,  C'wj,  and  add  ;  then  we  have 


dt 


dt 
db)g 


dto. 


i-i) 


(3) 


integrating,  we  have 

Jw,«  +  Bo}^^  +  Ou^i  =  hi; 

where  A'  and  k'^  are  tlie  constants  of  integration. 
Eliminating  Wj^  from  (3),  we  have 

A  {A  -  C)  w,2  +  B{B-  C)  WgJ  =  k^  -  6%2; 
and  c.3«  =  ^-  ^^ 


[^•^  _  /y/^a  _  A  (.1  _  //)  w,aj.     (5) 


Substituting  these  values  ol'wg  and  (.>j  in  the  lirst  of  ecjua- 
tions  (1),  we  have 


+ 


\A-C){A-B)^     a     _'t2-<',7tM 

i?C'  T'  ~yr(.i-r7)/ 


which  is  generally  an  elliptic  transcendent,  and  so  does  not 
admit  of  intcgnitidii  ill  finite  tt-rnis.  In  certain  particular 
cases  it  may  be  iiitcirrated.  which  will  give  the  value  of  <•), 
in  terms  of  /,  and  if  this  value  be  substituted  in  (4)  and  (5), 


60-4      Al'l'LIVATIO.S   OF  rUE   at:.\EIiAL    K<iLAriOA-S. 

the  values  of  Wj  and  Wj  in  terms  of  /  will  l)e  known,  iind 
thus,  in  these  eases,  the  problem  admits  of  complete  solu- 
tion. 

CoK. — Let  lOx,  u)y,  (.)«  be  the  axial  components  of  the 
initial  angular  veloeity  about  the  principal  axes  when 
/  =  0;  then  integrating  the  first  of  (^i),  and  taking  the 
limits  corresponding  to  /  and  0,  we  have 

Jwi^  +  Z?Wjj2  +  C'wj"  =  .Iw/  +  Bm.^  +  do?.       (7) 

Let  «,  (3,  y  be  the  direction-angles  of  the  instantaneous 
axis  at  the  time  t  relative  to  the  j)riMcipal  axes;  so  tliat,  if 
«■)  is  the  instantaneous  angular  velocity,  and  !;/(/•'  is  tlu' 
moment  of  inertia  relative  to  that  axis,  we  have  (Art.  ih^). 
w,  =  w  cos  ((,  (.)g  =:  w  cos  /J,  Wj  =  oj  cos  y,  which  sub- 
stituted in  (T),  gives 

Ai^/  +  i?w^8  +  C'w/  =  i,yi  {A  cos2  ,i  +  B  cos2  /3  +  C'cos^  y) 

=  w2lwr;-2  (Art.  t^'i,  Cor.) 

=  the  vis  viva  of  the  body; 

from  which  it  appears  that  the  ri><  viva  of  the  body  is  con- 
stant tliroiiijlioiit  tlw  whole  motion. 

Ukm.— An  api^licatioM  of  the  general  ecpuitions  of  rotatory 
motion  (Art.  357),  which  is  of  great  interest  and  impor- 
tance, is  that  of  the  rotatory  phenomena  of  the  earth  under 
the  action  of  the  attracting  forces  of  the  sun  and  the  moon, 
tlu'  rotation  being  considered  relative  to  the  centre  of 
gravity  and  an  axis  j)assing  through  it,  just  as  if  the  centre 
of  gravity  was  a  ti.xed  point  (Art.  ^49,  Sch.)  ;  and  the 
jir()l)lem  treated  as  purely  a  mathematical  one.  Also,  in 
addition   to  the  sun  and   the  moon,  the  problem   may   be 


r 


(iC. IT  toys. 

be  knuwu,  iiiul       * 
)f  complete  sjolu- 


nponeiits  of  the 

ipal    axes   when 

and  takiufj  tlie 


K^  +  Om?.      (7) 

he  iiistantaricoii!) 
axes  ;  so  that,  if 
and  !//(/•'  is  (lie 
have  (Art.  :^53). 
us  y,  which  siib- 


;os2  /3  +  C  C082  y) 
Cor.) 

;  body ; 

f  the  body  is  cou- 

itions  of  rotatory 
rest  and  impor- 
'  tlie  earth  under 
n  and  the  moon, 
)  tlie  centre  of 
t  as  if  the  centre 
Sell.)  ;  and  the 
1  one.  Also,  in 
iirobleni   niav   be 


EXAM  PLUS. 


505 


extended  so  as  to  include  the  action  of  all  the  other  bodies 
whose  influence  affects  the  motion  of  the  eartii's  rotation, 
lu  fact  the  investigation  of  the  motion  of  a  system  of  bodies 
in  space  miglit  bo  continued  at  great  length  ;  but  such 
investigations  would  l>e  clearly  l)eyond  the  limits  i)roposed 
in  this  treatise.  The  student  who  desires  to  continue  this 
interesting  subject,  is  referred  to  more  extended  works.* 

EXAMPLES. 

1.  A  hollow  sidierical  shell  is  filled  with  fluid,  and  rolls 
down  a  rough  inclined  plane;  determine  its  motion. 

Let  .\f  and  J/'  be  the  masses  of  the  shell  and  fluid 
respectively,  k  and  k'  their  radii  of  gyration  respectively 
about  a  diameter,  and  a  and  a'  the  radii  of  the  exterior  and 
interior  surfaces  of  the  shell  ;  then  using  the  same  nota- 
tion as  in  Art.  240,  we  have 


iP.r 


(M  +  M')  ;^^  =.(.»/+  J/')5r  sin  a  -  F. 


(1) 


As  the  sphericsd  shell  rotates  in  its  descent  down  the  plane, 
the  fluid  has  only  motion  of  translation;  so  that  the  equa- 
tion of  rotation  is 

.^/^•^|,-  =  Fa.  (2) 

Multiplying  (1)  by  a''  aiul  (2)  by  a,  aiul  adding,  we  have 

[{M  +  M')  a^  +  Mk^  '^  -  (Af  +  M')  a^(/  ain  «.    (3) 

If  the  interior  were  solul,  and  rigidly  joined  to  the  .shell, 
the  equation  of  motion  would  Ik; 


•  Sec  Price'a  Mcch'g,  Vol.  II,  PrattV  MccbV,  HouthV  KiBid  Dynamics,  La  I'lacc's 
V.ccauiquc  CMlcBtc,  etc. 


m 


506 


KXAMI'I.KS. 


(Px 


L(.)/n'-J/')«H.l/^^  +  .!/'/t-^j"^  =  (.l/+.J/')«^^8in«.  (4) 


Intcf>ratiiig  {.])  and  (4)  twice,  and  denoting  l)y  *■  and  ,v'  the 
Himces  tlirongli  wiiicii  the  centre  moves  during  tlie  time  / 
in  tiiese  two  eases  respectively,  we  liave 


s'  (.¥  +  M ')  d^  -Y  Ml^ 


(5) 


Ro  that  a  greater  space  is  described  by  the  sphere  which  lias 
the  lliiid  than  by  that  which  lias  the  solid  in  its  interior. 

If  the  densities  of  the  solid  and  the  fluid  are  the  same, 
we  have  fro.a  (5),  by  Art.  233,  Ex.  14, 


>'  ~  7«5"^2(?5"     (P"'^^''^  ^"'^l-  '^Iw'l's-,  Vol.  II,  p.  368). 


2.  A   homogeneous  sphere  rolls  down   within    a  rough 
spherical  bow! ;  it  is  re(juired  to  determine  the  motion. 

Let  n  l)e  the  radius  of  the  sphere,  and  h  the  radius  of  the 
bowl ;  and  let  us  suppose  the  sphere  to  be  placed  in  the 
bowl  at  rest.  Lo"^  OCQ  =  <p, 
Ql'A  =  e,  BCO  =  (c,  o)  = 
the  angular  velocity  of  the 
ball  about  an  axis  through  its 
centre  P,  h  =  the  correspond- 
ing radius  of  gyration  ;  0.]f  = 
X,  MP  =  y  \  m  —  Wxa  mass  of 
the  ball.     Then  Fig.ioi 


—  /?  sin  0  +  /"cos  <p\ 


(1) 


^11 


")a^g  sin  «.  (4) 

by  *■  imd  ,v'  1  ho 
ing  the  time  / 


k'^ 


(5) 


here  which  has 
its  interior. 
I  are  the  sunie, 


.1.  II,  p.  368). 


ithin   a  rough 
le  motion, 
e  radius  of  the 
placed  in  the 


I.IOI 


'ffl 


(1) 


(^) 


EXAMI'LES. 


mi^  77  =  aF. 

(It 


507 
(3) 


Also     X  =    {h  —  «i)  sin  <p  ;    y  =  f/  —  {/>  —  a)  cos  <p. 


dfi 


=  (b-a)  cos  <p  ;;^J  -  '/.  -  a)  sin  0  Q  ,       (4) 

(6) 


(p'l 
m  {/>  —  a)    .     ^=  F  —  mtj  sin  0. 


(7) 


Now  to  doterminc  the  angular  velocity  of  the  ball,  we  must 
estimate  tlie  angle  described  by  a  lixed  line  in  it,  as  I'A, 
from  a  line  fixed  in  direction,  a.s  PM,  and  tiie  ratio  of  the 
jnfinitesimal  increase  of  this  angle  to  that  of  the  time  will 
lie  the  angular  velocity  of  the  ball. 

_  tlMPA  _  (H>       dd 
•*•     '^  -      (H       -  (It  '^  (if 

Since  the  sphere  does  not  slide,  (tO  =  b((c  —  0) ; 

a  —  b  c/0^ 


w  = 


a      (It  ' 


du)  _  a  —  b  (Prji^ 
dt   ~  "  rt      dP  ' 


from  (.'5),  (7),  and  (8)  we  get 

(P<t> 


(*-«)^^"  =  --fi/siuc^; 


(8) 


(9) 


508  FXAMl'KES. 

,,  ,  /d(b\^       10,7  , 

•••    (*  —  «)  (,/[/  =  -y  (^^^  't'  —  ^^^  «)•  (1^) 

Substituting  (9)  in  (7)  we  have 

F  =  ^y/*//  sill  (/».  (11) 

Substituting  (4),  (!)),  (10),  (11)  in  (1)  we  have 

li  z=^'-^(l7  cos  4>  —  10  cos  «)  J 

tlioreforc  the  pressure  at  the  lowest  point 

=  '-„-  (1<  —  10  cos  u); 

and  the  pressure  of  tiie  ball  on  the  bowl  vanishes  wlten 

cos  0    :=:   i-^  COS  «. 

Cor.— If  tlic  l)all  rolls  ove:-  a  small  arc  at  the  lowest  part 
of  the  bowl,  so  that   «  and  D  are  always  small,  co-  »«,  and 

cos  (p  may  be  replaced  by  I  —  ^  -uid   1  —  —  re8])ectively ; 
and  from  (10)  we  have 


t; 


tliiis  (he  ball  comes  (o  rest  at  points  whoso  angular  distance 
IS  «  on  both  sides  of  (I,  the  lowest  point  of  the  iiowl  ;  and 
the  periodic  time  is 


—  af0 

;  {h  -  «)_ 

(«a  _  ,/,2)i  ~ 

'.      0   z=    «  COS 

:  [b  -  a] 

'[p'^-^l 


0. 


(10) 


(11) 


ve 


ishes  when 


Hie  lowest  piirf: 
mil,  0()>  «,  iiiid 

l>'  ■     , 

J  res])ectively ; 


iigular  (listaiicp 
the  bowl  ;  iiiiil 


A'AM.l/y'/,AW. 


500 


therefore  the  oscillations  are  ])errorme(l  in  the  same  time  as 
those  of  a  simple  pendulum  whose  length  is  }  {It  —  a), 
(Art.  l'.)4).     (Price's  Anal.  Mech's,  Vol.  II,  p.  369.) 

3.  A  homogeneous  sphere  has  an  angular  velocity  (.i 
about  its  diameter,  and  gradualh  contracts,  remaining 
constantly  homogeneous,  till  it  has  half  the  ori!j;inal 
diameter;  recpiired  the  tioal  angular  velocity.     ,l/(.v.   \m. 

4.  If  the  earth  were  a  honiosreneous  sphere,  at  what  point 
must  it  be  struck,  that  it  may  receive  its  present  velocity 
of  translatioi'  and  of  rotation,  tiie  former  being  (JISOOO  miles 
l)er  iiour  tiearly?     Ann.   '^4  miles  lunirly  from  the  centre. 

T).  A  homogeneous  sphere  rolls  dow.i  a  rough  inclined 
j)lane;  the  inclined  plane  rests  on  a  smooth  horizontal 
plane,  along  which  it  slides  by  reason  of  the  pressure  of  the 
sphere;  required  the  motions  of  the  inclined  plane  and  of 
ihe  centre  of  the  sphere. 

Let  in  =  the  mass  of  tlie  sphere, 
M  =  the  mass  of  the  inclined 
plane,  a  ==  the  radius  of  the  sphere, 
a  =  the  angle  of  the  inclined 
plane,  Q  its  apex  ;  0  the  place  of  Q 
when  /  =  0 ;  0'  the  point  on  the 
])lane  which  was  in  contact  with 
the  point  A  of  tlie  sphere  when 
/  =  0,  at  which  time  we  may  sup- 
])ose  all  to  be  at  rest ;  ACP  =  0,  the  angle  through  which 
the  sphere  has  revolved  in  the  time  t. 

Let  0  \w  the  origin,  and  let  the  horizontal  and  vertical 
lines  through  it  be  the  axes  of  r  and  // ;  OQ  —.  ,r'  ;  and  let 
(.r.  //)  (//,  k)  be  the  jdaces  of  the  centre  of  the  sphere  at  the 
times  /  =  /  and  /  ■—  0  respectively.  Then  the  e({uatioii8 
of  motion  of  the  sphere  are 

tPx 


Fig.  102 


m 


dP 


F  cos  «  —  R  sin 


a. 


^ 


510 


KXAMl'LEH. 


|ww' 


dp 


nF', 


and  the  equation  of  motion  of  tlie  i)hine  is 

(Px' 
^  ^^2  —  —  -Fcosfc  +  It  sin  «. 

From  the  geometry  we  have 

X  =.  h  -{-  x'  —  aO  cos  a, 

y  =.  k  —  ad  sin  o. 

From  these  equations  we  obtain 

,       m  cos  a    - 
tn  +  M 

—  5>w  sin  «  cog  «  ^/* 

~  7  {m  +'M )  —  5wj  cos^  «  *  T ' 


a;  =  A 


5^/sin  «  cos  a 


7(/«  +  M)  —  5m  cos^K     2' 


_  , 5  (?»  +  ii/)  sin*  rt         9^» 

^  "~  7  (m  +  i/ )  _  Ahj  cos*  «  ■ " y 

which  give  the  vahiea  of  a;  and  y  in  terms  of  t. 
Also  W3  obtain 

(m  +  Af)  {x  -  ft)  sin  a  ~  M  {y  —  k)  cos  «  =  0  ; 

which  is  the  equation  of  the  path  described  by  the  centre 
of  the  sphere  ;  and  therefore  this  path  is  a  straigiit  line. 

6.  A  heavy  solid  wheel  in  the  form  of  a  right  circular 
cylinder,  is  composed  of  two  substances,  whose  volumes  are 


,1 


mg. 


w. 


a 

J^. 

I  cos^ « 

a » 

fft' 

%' 

)s  rt  =  0  ; 

l).v  the  centre 
niiglit  lint". 

right  cirt'ulur 
se  volumes  are 


i 


KXAMl'LEH. 


fill 


equal,  iiiid  wliose  densities  are  />  and  p  ;  these  substances 
are  arranged  in  two  different  forms;  in  one  case,  that  whose 
di'nsitv  is  p  (iccn|>ies  tlie  central  part  of  the  wheel,  and  tiie 
other  is  placed  as  a  ring  round  it;  in  the  secontl  case,  the 
places  of  the  substances  are  interchanged  ;  /  and  /'  are  the 
times  in  whidi  the  wheels  roll  down  a  given  rough  inclined 
[ilane  from  rest;  show  that 

(i  :  /'•■*  : :  5/)  +  tp'  :  5p'  +  7p. 

7.  A  homogeneous  sphere  moves  down  a  rough  inclined 
plane,  whose  angle  of  inclinatiou  «  to  the  horizon  is  greater 
than  that  of  the  angle  of  friction  ;  it  is  recjuired  to  show  (1) 
that  the  sphere  will  roll  without  sliding  when  fi  is  equal  to 
or  greater  than  f  tan  «,  and  {'X)  that  it  will  slide  and  roll 
when  }i  is  less  than  f  tan  «,  where  /t  is  the  coeflicient  of 
friction. 

8.  In  the  lask  example  show  that  the  angular  velocity  of 
the  sphere  at  the  time  t  irom  rest  =  — '^ 1. 

y.  If  the  body  moving  down  the  plane  is  a  circular 
cylinder  of  radius  =  a,  with  its  axis  horizontal,  show  that 
tlie  body  will  slide  and  roll,  or  roll  only,  according  as  «  is 
greater  or  not  greater  than  tau"*  'i(i. 


ite 


